Samuël Borza
email: sborza@sissa.it

Mattia Magnabosco
email: mattia.magnabosco@maths.ox.ac.uk

Tommaso Rossi
email: tommaso.rossi@inria.fr

Kenshiro Tashiro
email: kenshiro.tashiro.b2@tohoku.ac.jp

Measure contraction property and curvature-dimension condition on sub-Finsler Heisenberg groups

In this paper, we investigate the validity of synthetic curvature-dimension bounds in the sub-Finsler Heisenberg group, equipped with a positive smooth measure. Firstly, we study the measure contraction property, in short MCP, proving that its validity depends on the norm generating the sub-Finsler structure. Indeed, we show that, if it is neither C 1 nor strongly convex, the associated Heisenberg group does not satisfy MCP(K, N ) for any pair of parameters K ∈ R and N ∈ (1, ∞). On the contrary, we prove that the sub-Finsler Heisenberg group, equipped with a C 1,1 and strongly convex norm, and with the Lebesgue measure, satisfies MCP(0, N ) for some N ∈ (1, ∞). Additionally, we provide a lower bound on the optimal dimensional parameter, and we also study the case of C 1 and strongly convex norms. Secondly, we address the validity of the curvature-dimension condition pioneered by Sturm and Lott-Villani, in short CD(K, N ). We show that the sub-Finsler Heisenberg group, equipped with a C 1 and strongly convex norm, and with a positive smooth measure, does not satisfy the CD(K, N ) condition for any pair of parameters K ∈ R and N ∈ (1, ∞). Combining this result with our findings regarding the measure contraction property, we conclude the failure of the CD condition in the Heisenberg group for every sub-Finsler structure.

Introduction

In the seminal contributions [Stu06b, Stu06a, LV09], Sturm and Lott-Villani introduced the celebrated curvature-dimension condition, in short CD(K, N ). This was the first successful notion of synthetic Ricci curvature bound in the non-smooth setting of metric measure spaces. The crucial observation was that a weighted Riemannian manifold has generalized Ricci curvature bounded below by K ∈ R and dimension bounded above by N ∈ (1, +∞] if and only if the so-called N -Rényi entropy functional is (K, N )-convex along Wasserstein geodesics, cf. [START_REF] Cordero-Erausquin | A Riemannian interpolation inequality à la Borell, Brascamp and Lieb[END_REF][START_REF] Von Renesse | Transport inequalities, gradient estimates, entropy, and Ricci curvature[END_REF].

While the former is a differential notion, the latter relies only on the underlying metric structure and on a reference measure, thus it can be promoted to a definition of synthetic curvaturedimension bound for metric measure spaces. Subsequently, it was showed in [START_REF] Ohta | Finsler interpolation inequalities[END_REF] that the same relation between the CD condition and the (flag) Ricci curvature holds in smooth Finsler manifolds.

While in Riemannian and Finsler geometry, the curvature-dimension condition à la Lott-Sturm-Villani is consistent with a lower bound on the Ricci curvature, the same does not hold in sub-Riemannian and sub-Finsler geometry. The latter are a broad generalization of Riemannian and Finsler geometry, where a smoothly varying norm is defined on a subset of preferred directions, called distribution. In a series of contributions [START_REF] Juillet | On a method to disprove generalized Brunn-Minkowski inequalities[END_REF][START_REF] Juillet | Sub-riemannian structures do not satisfy riemannian brunn-minkowski inequality[END_REF][START_REF] Magnabosco | Almost-Riemannian manifolds do not satisfy the curvature-dimension condition[END_REF][START_REF] Rizzi | Failure of curvature-dimension conditions on sub-Riemannian manifolds via tangent isometries[END_REF], it has been showed that the CD condition fails in sub-Riemannian geometry. More precisely, given a (truly) sub-Riemannian manifold M , equipped with a smooth positive measure m, the metric measure space (M, d SR , m) does not satisfy the CD(K, N ) condition, for any choice of parameters K ∈ R and N ∈ (1, +∞]. Note that, the positivity of the measure is not merely a technical assumption as there are examples of sub-Riemannian manifolds, equipped with smooth measures vanishing at some points, that satisfy the CD condition, see [START_REF] Pan | Examples of Ricci limit spaces with non-integer Hausdorff dimension[END_REF][START_REF] Pan | The Grushin hemisphere as a Ricci limit space with curvature ≥ 1[END_REF]. In addition, recent developments [START_REF] Borza | Measure contraction property, curvature exponent and geodesic dimension of sub-Finsler Heisenberg groups[END_REF][START_REF] Magnabosco | Failure of the curvature-dimension condition in sub-finsler manifolds[END_REF] have shown that the CD condition fails also in a large class of sub-Finsler manifolds, corroborating the general belief that it should fail in sub-Finsler geometry.

These results show that the classical CD condition is not suitable to study curvature in the setting of sub-Riemannian and sub-Finsler manifolds. Motivated by this, Barilari, Mondino and Rizzi [START_REF] Barilari | Unified synthetic Ricci curvature lower bounds for Riemannian and sub-Riemannian structures[END_REF] introduced and studied a generalized version of the CD condition, defined for gauged metric measure spaces. They showed in particular that any compact fat sub-Riemannian manifold, with a suitable gauge, satisfies their version of the CD condition.

A different classical curvature-dimension bound, which instead holds in some examples of sub-Riemannian and sub-Finsler manifolds, is the so-called measure contraction property, or MCP(K, N ) for brevity. This condition was introduced by Ohta in [START_REF] Ohta | On the measure contraction property of metric measure spaces[END_REF] and it prescribes a control (depending on K and N ) on the contraction rate of volumes along geodesics. Although the MCP(K, N ) condition is weaker than the CD(K, N ) condition, in the setting of (unweighted) Riemannian manifolds, it is equivalent to having Ricci curvature bounded below by K ∈ R and dimen-sion equal to N ∈ (1, +∞). As mentioned, the MCP condition holds in a large class of sub-Riemannian manifolds, including many Carnot groups, cf. [Rif13, Riz16, BR18, BR20, Bor22, NGZ23], and in the (sub-Finsler) ℓ p -Heisenberg group for 1 < p ≤ 2, as shown by the first-and fourthnamed authors in [START_REF] Borza | Measure contraction property, curvature exponent and geodesic dimension of sub-Finsler Heisenberg groups[END_REF]. In particular, for p = 2, they recovered a result of [START_REF] Juillet | On a method to disprove generalized Brunn-Minkowski inequalities[END_REF], where it was shown that the sub-Riemannian Heisenberg group, equipped with the Lebesgue measure L 3 , satisfies MCP(0, 5) with sharp constants.

In this paper, we complete the study initiated in [START_REF] Borza | Measure contraction property, curvature exponent and geodesic dimension of sub-Finsler Heisenberg groups[END_REF][START_REF] Magnabosco | Failure of the curvature-dimension condition in sub-finsler manifolds[END_REF], investigating whether the measure contraction property MCP(K, N ) holds in the sub-Finsler Heisenberg group, equipped with a general norm. The sub-Finsler Heisenberg group is a stratified real Lie group of dimension 3, where a norm • is defined on the first layer of its Lie algebra. In this setting, we prove both positive and negative results, showing that the validity of MCP(K, N ) depends on the smoothness and convexity properties of the reference norm • . This is particularly interesting if compared to what happens in R n (with the Lebesgue measure L n ), which satisfies CD(0, n), and thus MCP(0, n), if equipped with any distance induced by a norm, see the appendix of [START_REF] Villani | Optimal transport: Old and New[END_REF].

Firstly, we study in detail the geometry of the sub-Finsler Heisenberg group, making use of convex trigonometry (cf. Subsections 2.2 and 2.3), in the way that it was introduced in [START_REF] Lokutsievskiy | Convex trigonometry with applications to sub-Finsler geometry[END_REF]. Building upon [START_REF] Berestovskiȋ | Geodesics of nonholonomic left-invariant inner metrics on the Heisenberg group and isoperimetrics of the Minkowski plane[END_REF][START_REF] Lokutsievskiy | Explicit formulae for geodesics in left-invariant sub-Finsler problems on Heisenberg groups via convex trigonometry[END_REF], we provide a precise and general description of geodesics and their uniqueness: one of our main results (Proposition 3.5) highlights that uniqueness of geodesics depends on how "flat" the polar of the unit ball of the reference norm is and extends [START_REF] Breuillard | On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry[END_REF]. Secondly, this allows us to identify a sub-Finsler exponential map and study its regularity properties, in relation to the properties of • . The main observation is that the smoothness of the norm • influences the positivity of the Jacobian of the exponential map, while the convexity of

• determines its regularity. As the Jacobian of the exponential map controls the infinitesimal volume distortion of geodesics (cf. Subsection 3.2), we use it to develop many useful criteria to address the validity of the MCP condition in the sub-Finsler Heisenberg group. Our first result in this direction (Proposition 3.18) is valid for any norm and provides a general necessary condition for MCP to hold, in term of the Jacobian of the exponential map. Our second result (Proposition 3.20) shows that, if the reference norm • is C 1 and strictly convex, the same condition is also sufficient. Finally, a refined analysis of the differentiability properties of the Jacobian allows us to provide further differential characterizations for MCP, if the reference norm • is C 1 and strongly convex, cf. Corollary 3.22 and Proposition 3.23.

Relying on these criteria, we deduce our main result showing the failure of the measure contraction property.

Theorem 1.1. Let H be the sub-Finsler Heisenberg group, equipped with a norm • which is not C 1 or not strongly convex, and let m be a positive smooth measure on H. Then, the metric measure space (H, d, m) does not satisfy the measure contraction property MCP(K, N ) for every K ∈ R and N ∈ (1, ∞).

Remarkably, Theorem 1.1 shows that the properties of the reference norm • do not only influence the optimal constants K and N for which the measure contraction MCP(K, N ) holds, but the validity of the condition itself. The two main properties of • causing the failure of MCP are the non-smoothness and the lack of (strong) convexity and, notably, each of these two properties is reflected in a specific singular behavior of geodesics that we can exploit to show the failure of MCP, with two different strategies.

Indeed, if the reference norm • is not C 1 , we show that geodesics can branch, even though they are unique, cf. Theorem 4.1. This behavior, which was already observed in [START_REF] Magnabosco | Failure of the curvature-dimension condition in sub-finsler manifolds[END_REF] for strictly convex not C 1 norms, has independent interest, as examples of branching spaces usually occur when geodesics are not unique. Instead, when the reference norm • is not strongly convex, we take advantage of the loss of regularity of the Jacobian of the exponential map. The proof of this case is divided into two parts: the first part (Theorem 4.5) addresses the scenario where the norm is not even strictly convex, while the second part (Theorem 4.6) closes the remaining gap. More in details, when the norm is not strictly convex we exploit discontinuity points of the Jacobian and contradict the necessary condition of Proposition 3.18. While, if the norm is strictly but not strongly convex, we contradict the equivalent characterization of Proposition 3.20, exploiting the fact that the Jacobian, while continuous, fails to be Lipschitz. Subsequently, we investigate the validity of the measure contraction property in the sub-Finsler Heisenberg group, when the reference norm is at least C 1 and strongly convex. The central object of our analysis is the angle correspondence map C • : R → R, which represents the duality map from (R 2 , • * ) to (R 2 , • ), interpreted at the level of generalized angles, see Subsection 2.2 for a precise definition. According to classical convex geometry, the duality map corresponds to the differential of the dual norm and, therefore, the angle correspondence map C • has its regularity tied to the regularity of the norm • , see Proposition 2.16. In our analysis, the angle correspondence appears in the asymptotic expansion of the Jacobian of the exponential map. Hence, when the norm • is C 1,1 and strongly convex, we shall observe that the map C • is bi-Lipschitz and we obtain an important positive result (Corollary 5.12).

Theorem 1.2. Let H be the sub-Finsler Heisenberg group, equipped with a C 1,1 and strongly convex norm • , and let L 3 be the Lebesgue measure on H. Then, the metric measure space (H, d, L 3 ) satisfies the measure contraction property MCP(0, N ) for some N ∈ (1, ∞).

Note that, as the Heisenberg group admits a one-parameter family of dilations (cf. [START_REF] Le | A metric characterization of Carnot groups[END_REF]), by the scaling property of MCP, it is sufficient to investigate the validity of the measure contraction property with K = 0.

Combining Theorem 1.1 and Theorem 1.2, we obtain an almost complete picture describing the validity of MCP in the sub-Finsler Heisenberg group. However, these results do not cover the case when the norm • is strongly convex and C 1 but not C 1,1 . In Section 5, we identify different behaviors for this intermediate case, showing that MCP can both hold or fail. The findings of Section 5 supporting the validity of MCP can be summarized in the following theorem (cf. Theorem 5.9).

Theorem 1.3. Let H be the sub-Finsler Heisenberg group, equipped with a C 1 and strongly convex norm • and with the Lebesgue measure L 3 . If the derivative of C • is asymptotically and uniformly equivalent to a fractional polynomial at its zero points, see (68) and (69), then (H, d, L 3 ) satisfies the MCP(0, N ) for some N ∈ (1, ∞).

The proof of Theorem 1.3 hinges upon the characterization of MCP given in Proposition 3.22 and a novel Taylor expansion of the Jacobian of the exponential map with integral remainder (see Proposition 5.6). This theorem generalizes the validity of MCP for ℓ p -norm with p ∈ (1, 2) that was obtained in [START_REF] Borza | Measure contraction property, curvature exponent and geodesic dimension of sub-Finsler Heisenberg groups[END_REF]. We remark that we indeed obtain Theorem 1.2 as a corollary of Theorem 1.3.

On the one hand, the sufficient conditions identified in Theorem 1.3 are not necessary. Indeed, we provide an example (cf. Example 5.16) of sub-Finsler Heisenberg group, equipped with a C 1 and strongly convex norm • , satisfying the MCP condition, but where C ′ • does not behave as a fractional polynomial. On the other hand, as displayed by Example 5.17, removing the uniformity assumption may lead to the failure of the MCP condition.

An additional relevant byproduct of our study is a lower bound on the so-called curvature exponent N curv of the sub-Finsler Heisenberg group. The curvature exponent is the minimal parameter N such that H (equipped with the Lebesgue measure L 3 ) satisfies MCP(0, N ). To get an estimate of the curvature exponent, we need a slightly stronger assumption on C • (cf. Theorem 5.10).

Theorem 1.4. Let H be the sub-Finsler Heisenberg group, equipped with a C 1 and strongly convex norm • and with the Lebesgue measure L 3 . If the derivative of C • has maximal fractional order s ≥ 0, see (70), then (H, d, L 3 ) satisfies the MCP(0, N ) for some N ∈ (1, ∞), and N curv ≥ 2s + 5.

An important consequence of Theorem 1.4 is that if the reference norm is C 2 and strongly convex, then the curvature exponent N curv ≥ 5 (cf. Corollary 5.12). However, the lower bound 5 is not sharp for a general C 2 norm. Indeed, we observe that there is a C 2 and strongly convex norm such that N curv > 5 (cf. Example 5.14). These facts lead us to formulate the following conjecture.

Conjecture 1.5. The metric measure space (H, d, L 3 ) satisfies MCP(0, 5) if and only if the reference norm is the ℓ 2 -norm, that is to say (H, d) is the sub-Riemannian Heisenberg group.

In the last section of this paper we prove the failure of the CD condition in the sub-Finsler Heisenberg group, equipped with a C 1 strongly convex norm • and with a smooth measure m (cf. Theorem 6.3). In light of Theorems 1.2 and 1.3, this is a completely non-trivial result as the weaker MCP condition may hold in this case. Our argument is a substantial refinement of the one presented in [START_REF] Magnabosco | Failure of the curvature-dimension condition in sub-finsler manifolds[END_REF] for C 1,1 and strictly convex norms, based on an improved analysis of the correspondence map C • . Combining the findings of Theorem 6.3 with Theorem 1.1, we obtain the following negative result, valid for any reference norm.

Theorem 1.6. Let H be the sub-Finsler Heisenberg group, equipped with a norm • and with a positive smooth measure m. Then, the metric measure space (H, d, m) does not satisfy the CD(K, N ) condition, for every K ∈ R and N ∈ (1, ∞).

The sub-Finsler Heisenberg groups are the unique (up to isometries) sub-Finsler Carnot groups with Hausdorff dimension less than 5 (or with topological dimension less than or equal to 3), see [ABB20, Def. 10.3] for a precise definition of Carnot group. Therefore, Theorem 1.6 corroborates the following conjecture, already formulated in [START_REF] Magnabosco | Failure of the curvature-dimension condition in sub-finsler manifolds[END_REF].

Conjecture 1.7. Let G be a sub-Finsler Carnot group, endowed with a positive smooth measure m. Then, the metric measure space (G, d SF , m) does not satisfy the CD(K, N ) condition for any K ∈ R and N ∈ (1, ∞).

Our interest in Carnot groups stems from the fact that they are the only metric spaces that are locally compact, geodesic, isometrically homogeneous and self-similar (i.e. admitting a dilation), cf. [START_REF] Le | A metric characterization of Carnot groups[END_REF]. According to this property, sub-Finsler Carnot groups naturally arise as (unique) metric tangents of metric measure spaces, as showed in [START_REF] Le | Metric spaces with unique tangents[END_REF]. As the metric measure tangents of a CD(K, N ) space are CD(0, N ), the study of the CD(K, N ) condition in sub-Finsler Carnot groups, and especially the validity of Conjecture 1.7, has the potential to provide deep insights on the structure of tangents of CD(K, N ) spaces. This could be of significant interest, particularly in connection with Bate's recent work [START_REF] Bate | Characterising rectifiable metric spaces using tangent spaces[END_REF], which establishes a criterion for rectifiability in metric measure spaces, based on the structure of metric tangents. from the ANR-DFG project "CoRoMo" (ANR-22-CE92-0077-01). K.T. is partially supported by JSPS KAKENHI grant numbers 18K03298, 19H01786, 23K03104.

Preliminaries

2.1 The CD(K, N ) and the MCP(K, N ) conditions A metric measure space is a triple (X, d, m) where (X, d) is a complete and separable metric space and m is a locally finite Borel measure on it. In the following, we denote by C([0, 1], X) the space of continuous curves from [0, 1] to X. For every t ∈ [0, 1] we call e t : C([0, 1], X) → X the evaluation map, i.e. e t (γ

) := γ(t). A curve γ ∈ C([0, 1], X) is said to be a geodesic if d(γ(s), γ(t)) = |t -s| • d(γ(0), γ(1)) for every s, t ∈ [0, 1].
We denote by Geo(X) the space of all geodesics on (X, d). The metric space (X, d) is said to be geodesic if every pair of points x, y ∈ X can be connected with a curve γ ∈ Geo(X). We denote by P(X) the set of Borel probability measures on X and by P 2 (X) ⊂ P(X) the set of those having finite second moment. We endow the space P 2 (X) with the Wasserstein distance W 2 , defined by

W 2 2 (µ 0 , µ 1 ) := inf π∈Adm(µ 0 ,µ 1 ) ˆd2 (x, y) dπ(x, y),
where Adm(µ 0 , µ 1 ) is the set of all admissible transport plans between µ 0 and µ 1 , namely all the measures π ∈ P(X × X) such that (p 1 ) ♯ π = µ 0 and (p 2 ) ♯ π = µ 1 , where p i , for i = 1, 2, is the projection onto the i-th factor. The metric space (P 2 (X), W 2 ) is itself complete and separable, moreover, if (X, d) is geodesic, then (P 2 (X), W 2 ) is geodesic as well. In this case, every geodesic (µ t ) t∈[0,1] in (P 2 (X), W 2 ) can be represented with a measure η ∈ P(Geo(X)), i.e. µ t = (e t ) # η.

The CD(K, N ) condition. We present the curvature-dimension condition, or CD(K, N ) for brevity, firstly introduced by Sturm and Lott-Villani in [START_REF] Sturm | On the geometry of metric measure spaces[END_REF][START_REF] Sturm | On the geometry of metric measure spaces[END_REF][START_REF] Lott | Ricci curvature for metric-measure spaces via optimal transport[END_REF]. For every K ∈ R, N ∈ (1, ∞) and t ∈ [0, 1], the distortion coefficients are the functions:

τ (t) K,N (θ) := t 1 N σ (t) K,N -1 (θ) 1-1 N , ∀ θ ≥ 0 where σ (t) K,N (θ) :=                    +∞ if N π 2 ≤ Kθ 2 , sin(tθ K/N ) sin(θ K/N ) if 0 < Kθ 2 < N π 2 , t if K = 0, sinh(tθ -K/N ) sinh(θ -K/N ) if K < 0.
Definition 2.1 (CD(K, N ) condition). A metric measure space (X, d, m) is said to be a CD(K, N ) space (or to satisfy the CD(K, N ) condition) if for every pair of measures µ 0 = ρ 0 m, µ 1 = ρ 1 m ∈ P 2 (X), absolutely continuous with respect to m, there exists a W 2 -geodesic (µ t ) t∈[0,1] connecting them and induced by η ∈ P(Geo(X)), such that µ t = ρ t m ≪ m for every t ∈ [0, 1] and the following inequality holds for every N ′ ≥ N and every t ∈ [0, 1]

ˆX ρ 1-1 N ′ t dm ≥ ˆX×X τ (1-t) K,N ′ d(x, y) ρ 0 (x) -1 N ′ + τ (t) K,N ′ d(x, y) ρ 1 (y) -1 N ′ dπ(x, y),
where π = (e 0 , e 1 ) # η.

The Brunn-Minkowski inequality. One of the main merits of the CD(K, N ) condition is that it is sufficient to deduce geometric and functional inequalities that hold in the smooth setting. An example, which is particularly relevant to this paper, is the so-called Brunn-Minkowski inequality, whose definition in a metric measure space requires the notion of midpoints.

Definition 2.2 (Midpoints). Let (X, d) be a metric space and let A, B ⊂ X be two Borel subsets. Then for t ∈ (0, 1), we define the set of t-midpoints between A and B as

M t (A, B) := {x ∈ X : x = γ(t) , γ ∈ Geo(X) , γ(0) ∈ A and γ(1) ∈ B} .
Definition 2.3 (Brunn-Minkowski inequality). Given K ∈ R and N ∈ (1, ∞), we say that a metric measure space (X, d, m) satisfies the Brunn-Minkowski inequality BM(K, N ) if, for every nonempty A, B ⊂ spt(m) Borel subsets and every t ∈ (0, 1), we have

m M t (A, B) 1 N ≥ τ (1-t) K,N (Θ(A, B)) • m(A) 1 N + τ (t) K,N (Θ(A, B)) • m(B) 1 N ,
where

Θ(A, B) :=    inf x∈A, y∈B d(x, y) if K ≥ 0 , sup x∈A, y∈B d(x, y) if K < 0 .
As already mentioned, the Brunn-Minkowski inequality is a consequence of the CD(K, N ) condition, in particular we have that

CD(K, N ) =⇒ BM(K, N ),
for every K ∈ R and every N ∈ (1, ∞). In Section 6, we will prove the failure of the CD(K, N ) condition for every choice of the parameters K ∈ R and N ∈ (1, ∞), by contradicting the Brunn-Minkowski inequality BM(K, N ). A priori, this is a stronger result than the one stated in Theorem 1.6, as in principle the Brunn-Minkowski inequality is weaker than the CD(K, N ) condition. However, recent developments (cf. [START_REF] Magnabosco | The Brunn-Minkowski inequality implies the CD condition in weighted Riemannian manifolds[END_REF][START_REF] Magnabosco | The strong Brunn-Minkowski inequality and its equivalence with the CD condition[END_REF]) suggest that the Brunn-Minkowski BM(K, N ) could be equivalent to the CD(K, N ) condition in a wide class of metric measure spaces.

The MCP(K, N ) condition. A way to relax the condition of CD(K, N ) involves requiring it only when the first marginal degenerates to δ x , a delta-measure at x ∈ spt(m), and the second marginal is m| A m(A) , for some Borel set A ⊂ X with 0 < m(A) < ∞. This is the idea behind behind the so-called measure contraction property, introduced by Ohta in [START_REF] Ohta | On the measure contraction property of metric measure spaces[END_REF].

Definition 2.4 (MCP(K, N ) condition). Given K ∈ R and N ∈ (1, ∞), a metric measure space (X, d, m) is said to satisfy the measure contraction property MCP(K, N ) if for every x ∈ spt(m) and every Borel set A ⊂ X with 0 < m(A) < ∞, there exists a W 2 -geodesic induced by η ∈ P(Geo(X)) connecting δ x and m| A m(A) such that, for every t ∈ [0, 1],

1 m(A) m ≥ (e t ) # τ (t) K,N d(γ(0), γ(1)) N η(dγ) . (1)
The MCP(K, N ) condition is weaker than the CD(K, N ) one, i.e.

CD(K, N ) =⇒ MCP(K, N ),
for every K ∈ R and every N ∈ (1, ∞), cf. [CM21, Lem. 6.13].

Remark 2.5. Let us recall a useful equivalent formulation of the inequality (1), which holds whenever geodesics are unique, we refer the reader to [Oht07, Lem. 2.3] for further details. Consider x ∈ spt(m) and a Borel set A ⊂ X with 0 < m(A) < ∞. Assume that for every y ∈ A, there exists a unique geodesic γ x,y : [0, 1] → X joining x and y. Then, (1) is verified for the marginals δ x and m| A m(A) if and only if

m M t ({x}, A ′ )) ≥ ˆA′ τ (t) K,N (d(x, y)) N dm(y), for any Borel set A ′ ⊂ A. (2)
We recall below the definition of the curvature exponent.

Definition 2.6 (Curvature exponent). Let (X, d, m) be a metric measure space satisfying the MCP(0, N ) for some N ∈ (1, +∞). The curvature exponent of X is defined as

N curv := inf{N ∈ (1, +∞) : (X, d, m) is MCP(0, N )}.
Scaling and stability properties. The CD(K, N ) condition and the measure contraction property MCP(K, N ) enjoy several properties that validate them as synthetic curvature dimension bounds. Among them, we only mention the ones necessary for our purposes:

• scaling property (cf. [START_REF] Sturm | On the geometry of metric measure spaces[END_REF]): If (X, d, m) is a CD(K, N ) (resp. MCP(K, N )) space, for every α, β > 0 the scaled space (X, αd, βm) is a CD(α -2 K, N ) (resp. MCP(α -2 K, N )) space.

• pmGH-stability (cf. [START_REF] Gigli | Convergence of pointed noncompact metric measure spaces and stability of Ricci curvature bounds and heat flows[END_REF]) Let {(X n , d n , m n , p n )} n∈N be a sequence of pointed metric measure spaces (i.e. metric measure spaces with a distinguished point) converging to

(X ∞ , d ∞ , m ∞ , p ∞ ) in the pointed measured Gromov-Hausdorff convergence. Assume that for every n ∈ N, (X n , d n , m n ) is a CD(K n , N n ) (resp. MCP(K n , N n )) space, for two sequences (K n ) n∈N ⊂ R and (N n ) n∈N ⊂ (1, ∞) converging to K ∞ ∈ R and N ∞ ∈ (1, ∞), respectively. Then, the limit space (X ∞ , d ∞ , m ∞ ) is a CD(K ∞ , N ∞ ) (resp. MCP(K ∞ , N ∞ )) space.

Convex trigonometry

In this section, we recall the definition and main properties of the convex trigonometric functions, firstly introduced in [START_REF] Lokutsievskiy | Convex trigonometry with applications to sub-Finsler geometry[END_REF]. Let Ω ⊂ R 2 be a convex, compact set, such that O := (0, 0) ∈ Int(Ω) and denote by π Ω its surface area.

Definition 2.7 (Convex trigonometric functions). Let θ ∈ R denote a generalized angle. If 0 ≤ θ < 2π Ω define P θ as the point on the boundary of Ω, such that the area of the sector of Ω between the rays Ox and OP θ is 1 2 θ (see Figure 1). Moreover, define sin Ω (θ) and cos Ω (θ) as the coordinates of the point P θ , i.e.

P θ = cos Ω (θ), sin Ω (θ) .

Finally, extend these trigonometric functions outside the interval [0, 2π Ω ) by periodicity (of period 2π Ω ), so that for every k ∈ Z cos Ω (θ) = cos Ω (θ + 2kπ Ω ), sin Ω (θ) = sin Ω (θ + 2kπ Ω ) and P θ = P θ+2kπ Ω .

In particular, the maps P, sin Ω , cos Ω are well-defined on the quotient R/2π Ω Z.

Observe that by definition sin Ω (0) = 0 and that when Ω is the Euclidean unit ball we recover the classical trigonometric functions.

Lemma 2.8. The map P : R/2π Ω Z → ∂Ω ⊂ R 2 that associate to the angle θ the vector P θ , is bi-Lipschitz.

Proof. First of all, observe that, by convexity of Ω, the map P is bijective and thus invertible. We now prove that P is Lipschitz. By compactness of ∂Ω, we can find ε > 0 and a positive constant

K such that, if θ 1 , θ 2 ∈ R with 0 < |θ 1 -θ 2 | < ε, we have d eu (O, l θ 1 ,θ 2 ) ≥ K,
where l θ 1 ,θ 2 is the line joining P θ 1 and P θ 2 . Then, given any

θ 1 , θ 2 ∈ R such that 0 < |θ 1 -θ 2 | < ε, we can deduce that 1 2 |θ 2 -θ 1 | ≥ L 2 △P θ 1 OP θ 2 ≥ 1 2 K • P θ 2 -P θ 1 eu ,
where △P θ 1 OP θ 2 denotes the triangle of vertices P θ 1 , O and P θ 2 . This proves that P is locally K-Lipschitz everywhere, and thus K-Lipschitz, since R/2π Ω Z is compact.

We are left to show that

P -1 is Lipschitz. Let θ 1 = θ 2 ∈ R such that ∠P θ 1 OP θ 2 < π 2
, where ∠P θ 1 OP θ 2 is the Euclidean angle between OP θ 1 and OP θ 2 . In this case, we consider the quantities r := min{ x eu : x ∈ ∂Ω} and R := max{ x eu : x ∈ ∂Ω}, and observe that the section of Ω between the rays OP θ 1 and OP θ 2 is contained in the triangle 2R r • △P θ 1 OP θ 2 . In fact, every point in the line segment joining 2R r P θ 1 and 2R r P θ 2 has Euclidean norm strictly bigger than R. Hence, we deduce that, for every

θ 1 = θ 2 ∈ R such that ∠P θ 1 OP θ 2 < π 2 , we have 1 2 |θ 2 -θ 1 | ≤ L 2 2R r • △P θ 1 OP θ 2 = 4R 2 r 2 L 2 △P θ 1 OP θ 2 ≤ 2R 2 r 2 R • P θ 2 -P θ 1 eu .
Therefore, we can conclude that the map P -1 is locally 4R 3 r 2 -Lipschitz, thus it is also 4R 3 r 2 -Lipschitz.

Remark 2.9. Since the projection R 2 ∋ (x, y) → x ∈ R (resp. R 2 ∋ (x, y) → y ∈ R) is Lipschitz, we deduce that the generalized trigonometric function cos Ω (resp. sin Ω ) is Lipschitz continuous.

sin Consider now the polar set:

Ω (θ) cos Ω (θ) O 1 2 θ Ω P θ
Ω • := {p ∈ R 2 : p, x ≤ 1 for every x ∈ Ω},
which is itself a convex, compact set such that O ∈ Int(Ω • ). Therefore, we can consider the trigonometric functions sin Ω • and cos Ω • . Observe that, by definition of polar set, it holds that cos Ω (θ) cos Ω • (ϕ) + sin Ω (θ) sin Ω • (ϕ) ≤ 1, for every θ, ϕ ∈ R.

Definition 2.10 (Correspondence). We say that two angles θ, ϕ ∈ R correspond to each other and write θ Ω ← → ϕ if the vector Q ϕ := (cos Ω • (ϕ), sin Ω • (ϕ)) determines a half-plane containing Ω (see Figure 2). By the bipolar theorem [Roc70, Thm. 14.5], it holds that Ω

θ Ω ← → ϕ if and only if cos Ω (θ) cos Ω • (ϕ) + sin Ω (θ) sin Ω • (ϕ) = 1. ( 3
)
The correspondence θ Ω ← → ϕ is not one-to-one in general, in fact if the boundary of Ω has a corner at the point P θ , the angle θ corresponds to an interval of angles (in every period). Nonetheless, we can define a monotone multi-valued map C • that maps an angle θ to the maximal closed interval containing angles corresponding to θ i.e. θ Ω ← → C • (θ). This function has the following periodicity property:

C • (θ + 2π Ω k) = C • (θ) + 2π Ω • k for every k ∈ Z, (4)
where π Ω • denotes the surface area of Ω • . Analogously, we can define the map C • associated to the correspondence ϕ

Ω •
←→ θ, and it satisfies an analogue of (4). Note that C • and C • are monotone and multi-valued maps, thus their composition is monotone and multi-valued as well. In particular, C • • C • (θ) is an interval containing θ, according to Proposition 2.11. The analogous property holds for the composition C • • C • (ϕ), and, for the sequel, we define the functions δ ± : R/2π

Ω • Z → [0, ∞) so that C • • C • (ϕ) = [ϕ -δ -(ϕ), ϕ + δ + (ϕ)].
(5)

Observe that the set

{Q ψ : ψ ∈ [ϕ -δ -(ϕ), ϕ + δ + (ϕ)]} is the maximal segment of ∂Ω • containing Q ϕ .
Proposition 2.12. Let Ω ⊂ R 2 as above. The trigonometric functions sin Ω and cos Ω are differentiable almost everywhere (cf. Remark 2.9). Their differentiability points coincide with the angles θ where C • is single-valued and it holds that

sin ′ Ω (θ) = cos Ω • (C • (θ)) and cos ′ Ω (θ) = -sin Ω • (C • (θ)).
Naturally, the analogous result holds for the trigonometric functions sin Ω • and cos Ω • .

According to the previous proposition, letting D 0 ⊂ R/2π Ω • Z be the set of differentiability points of sin Ω • and cos Ω • , D 0 is a L 1 -full-measure set and corresponds to the set where C • is a single-valued map. From now on, in order to ease the notation, we will sometimes use the shorthand:

ϕ • = C • (ϕ),
for the angles ϕ ∈ D 0 , where the correspondence map C • is single-valued.

Convex trigonometry associated with a norm

In the following, we study the convex trigonometry associated with the closed unit ball of a norm • on R 2 , i.e. Ω := B • 1 (0). In this case, the polar set Ω • is the closed unit ball B • * 1 (0) of the dual norm • * .

We say that a function f : R 2 → R is strictly convex if, for any x, y ∈ R 2 such that x = y,

f (tx + (1 -t)y) < tf (x) + (1 -t)f (y), ∀ t ∈ (0, 1).
Furthermore, let • : R 2 → R ≥0 be a norm, then we say that f is strongly convex with respect to • if there exists α > 0 such that, for every x, y ∈ R n ,

f (tx + (1 -t)y) ≤ tf (x) + (1 -t)f (y) - α 2 t(1 -t) x -y 2 , ∀ t ∈ [0, 1].
Let • : R 2 → R ≥0 be a norm on R 2 and define f Ω : R 2 → R ≥0 to be the function given by f Ω (x) := 1 2 x 2 . Similarly, we define f Proposition 2.14. Let • : R 2 → R >0 be a norm on R 2 , and let • * be its dual norm, then:

Ω • : R 2 → R ≥0 as f Ω • (x) := 1 2 x 2 * ,
(i) • is a strictly convex norm if and only if • * is a C 1 norm, i.e. f Ω • is C 1 ; (ii) • is a strongly convex norm if and only if • * is a C 1,1 norm, i.e. f Ω • is C 1,1 .
Furthermore, we can relate the regularity of the angle correspondence C • with the regularity of the norm. Define the map Q : R/2π Ω • Z → ∂Ω • ⊂ R 2 that associate to the angle ϕ the vector Q ϕ (this is the analogous map of the one defined in Lemma 2.8 for Ω).

Lemma 2.15. Let • be a norm and let

ϕ ∈ R/2π Ω • Z be an angle such that Q ϕ ∈ ∂Ω • is a differentiability point of • * . Then, the angle correspondence map C • from Ω • to Ω satisfies C • (ϕ) = P -1 • d • * • Q(ϕ) = P -1 • d Qϕ • * .
Proof. According to the Pythagorean identity (3), ϕ

Ω • ←→ θ = C • (ϕ) if and only if P θ is a dual vector of Q ϕ . Thus, if Q ϕ is a differentiability point of • * , [MR23b, Lem. 3.6] ensures that (cos Ω (θ), sin Ω (θ)) = P θ = d Qϕ • * .
The thesis follows from the definition of the maps P, Q, recalling also that P is invertible.

Proposition 2.16. Let • be a norm on R 2 and C • be the angle correspondence map from Ω • to Ω, then: (iii) The thesis can be proven similarly to item (ii), observing that the map

(i) • is a C 1 norm if and only if C • is strictly increasing, (ii) • is a strictly convex norm if
C • = P -1 • d • * • Q is Lipschitz if and only if d • * is Lipschitz, as a consequence of Lemma 2.8.
Finally, since in this case the set Ω is symmetric with respect to the origin, we have the following identities for the generalized trigonometric functions

cos Ω • (ϕ ± π Ω • ) = -cos Ω • (ϕ) and sin Ω • (ϕ ± π Ω • ) = -sin Ω • (ϕ) ∀ϕ ∈ R. (6)
The same symmetry guarantees the following property of the correspondence C • :

C • (ϕ ± π Ω • ) = C • (ϕ) ± π Ω . (7)
Remark 2.17. All the properties listed above, namely Lemma 2.15, Proposition 2.16, as well as the analogous of ( 6) and ( 7), hold for cos Ω , sin Ω and C • .

The sub-Finsler geometry of the Heisenberg group

We present here the sub-Finsler Heisenberg group and study its geodesics. Let us consider the Lie group M = R 3 , equipped with the non-commutative group law, defined by

(x, y, z) ⋆ (x ′ , y ′ , z ′ ) = x + x ′ , y + y ′ , z + z ′ + 1 2 (xy ′ -x ′ y) , ∀ (x, y, z), (x ′ , y ′ , z ′ ) ∈ R 3 ,
with identity element e = (0, 0, 0). We define the left-invariant vector fields

X 1 := ∂ x - y 2 ∂ z , X 2 := ∂ y + x 2 ∂ z .
The associated distribution of rank 2 is D := span{X 1 , X 2 }. It can be easily seen that D is bracket-generating. Then, letting • : R 2 → R ≥0 be a norm, the sub-Finsler Heisenberg group H is the Lie group M equipped with the sub-Finsler structure (D, • ). For further details on sub-Finsler geometry, we refer to [MR23b, Sec. 2.2]. We define the associated left-invariant norm on D as

v D := (u 1 , u 2 ) , for every v = u 1 X 1 + u 2 X 2 ∈ D.
A curve γ : [0, 1] → H is admissible if its velocity γ(t) exists almost everywhere and there exists

a function u = (u 1 , u 2 ) ∈ L 2 ([0, 1]; R 2 ) such that γ(t) = u 1 (t)X 1 (γ(t)) + u 2 (t)X 2 (γ(t)) ∈ D γ(t) , for a.e. t ∈ [0, 1].
The function u is called the control. We define the length of an admissible curve:

ℓ(γ) := ˆ1 0 γ(t) D dt ∈ [0, ∞).
For every couple of points q 0 , q 1 ∈ M , define the sub-Finsler distance between them as d(q 0 , q 1 ) := inf {ℓ(γ) : γ admissible, γ(0) = q 0 and γ(1) = q 1 } .

We recall that the Chow-Rashevskii Theorem ensures that the sub-Finsler distance on H is finite, continuous on and the induced topology is the manifold one.

Remark 3.1. Since both the norm and the distribution are left-invariant, the left-translations defined by

L p : H → H; L p (q) := p ⋆ q, (8)
are isometries for every p ∈ H.

Definition 3.2. Let H be the Heisenberg group. We say that a Borel measure m on H is smooth if it is absolutely continuous with respect the Lebesgue measure L 3 with a smooth and strictly positive density.

Geodesics in the Heisenberg group

In the sub-Finsler Heisenberg group, the geodesics were originally studied in [START_REF] Busemann | The isoperimetric problem in the Minkowski plane[END_REF] and [START_REF] Berestovskiȋ | Geodesics of nonholonomic left-invariant inner metrics on the Heisenberg group and isoperimetrics of the Minkowski plane[END_REF] for the three-dimensional case and in [START_REF] Lokutsievskiy | Explicit formulae for geodesics in left-invariant sub-Finsler problems on Heisenberg groups via convex trigonometry[END_REF] for general left-invariant structures on higherdimensional Heisenberg groups. Now, we define the map G t which plays the role of a sub-Finsler exponential map from the origin at time t, as justified by Propositions 3.4 and 3.5 below.

Definition 3.3. Let H be the sub-Finsler Heisenberg group, equipped with a norm • , let Ω be the associated closed unit ball and Ω • its polar. Let

U := R >0 × R/2π Ω • Z × {(-2π Ω • , 2π Ω • ) \ {0}}.
For every t ∈ R, we define the continuous mapping

G t : U → H such that for any (r, ϕ, ω) ∈ U , G t (r, ϕ, ω) := (x t , y t , z t ), where            x t (r, ϕ, ω) = r ω (sin Ω • (ϕ + ωt) -sin Ω • (ϕ)) , y t (r, ϕ, ω) = - r ω (cos Ω • (ϕ + ωt) -cos Ω • (ϕ)) , z t (r, ϕ, ω) = r 2 2ω 2 (ωt + cos Ω • (ϕ + ωt) sin Ω • (ϕ) -sin Ω • (ϕ + ωt) cos Ω • (ϕ)) . (9)
We stress that the domain U does not contain ω = 0, ±2π Ω • .

The curve [0, 1] ∋ t → G t (r, ϕ, ω) ∈ H satisfies the Pontryagin maximum principle, cf. [Lok21, Thm. 4], thus, as a consequence of [Ber94, Thm. 1], we deduce the following result.

Proposition 3.4. The curve γ (r,ϕ,ω) : [0, 1] → H, defined as γ (r,ϕ,ω) (t) := G t (r, ϕ, ω) is a geodesic between its endpoints e = γ (r,ϕ,ω) (0) and γ (r,ϕ,ω) (1). Proposition 3.5. Let H be the sub-Finsler Heisenberg group, equipped with a norm • and let

R := {(r, ϕ, ω) ∈ U : -2π Ω • + δ + (ϕ) < ω < 2π Ω • -δ -(ϕ)},
where δ ± are defined as in (5).

For (r, ϕ, ω) ∈ R, the curve γ (r,ϕ,ω) : [0, 1] → H; γ (r,ϕ,ω) (t) = G t (r, ϕ, ω) is the unique geodesic between its endpoints. If the norm • is C 1 , then R = U . While, if the norm • is strictly convex, then G t is continuously extended to ω = 0 by      x t (r, ϕ, 0) = (r cos Ω (ϕ • )) t, y t (r, ϕ, 0) = (r sin Ω (ϕ • )) t, z t (r, ϕ, 0) = 0,
where we use the shorthand ϕ

• := C • (ϕ). Finally, if the norm • is C 1 and strictly convex, then G 1 : U → {(x, y, z) ∈ H : z = 0, (x, y) = (0, 0)} is a homeomorphism.
Notation 3.6. We fix two important notations:

(i) Denote by Ω • (r,ϕ,ω) the following transformation of Ω • = B • * 1 (0): Ω • (r,ϕ,ω) := R -π/2 r ω (Ω • -(cos Ω • (ϕ), sin Ω • (ϕ))) ,
where R -π/2 is counter-clockwise rotation in the plane of angle -π/2.

(ii) For a continuous curve γ = (x, y, z) : [0, 1] → H and for every t ∈ [0, 1], we denote by A t (γ) the oriented area that is swept by the vector joining (0, 0) with (x(s), y(s)), for s ∈ [0, t].

The proof of Proposition 3.5 is based on the following results.

Proposition 3.7 ([Ber94, Thm. 1]). Let H be the sub-Finsler Heisenberg group, equipped with a norm • . Consider the curve γ (r,ϕ,ω) for some (r, ϕ, ω) ∈ U . Then, the curve t → (x t (r, ϕ, ω), y t (r, ϕ, ω)), which is the projection of γ (r,ϕ,ω) onto {z = 0}, belongs to the boundary of Ω • (r,ϕ,ω) . Moreover, for every t ∈ 0, 2π Ω • |ω| , we have z(t) = A t (γ).

Proposition 3.8 ([Nos08, Thm. 6]). There exists a continuous function µ : R 2 → R ≥0 , which is 2-homogeneous, i.e. µ(λp) = λ 2 µ(p) for every λ ∈ R, such that if |z| > µ(x, y) then any geodesic connecting the origin e with the point (x, y, z) projects to a subpath of a unique (up to translations) isoperimetric profile on the plane {z = 0}.

Remark 3.9. According to [Nos08, Thm. 6], for every p = 0 ∈ R 2 the quantity µ(p) is the infimum of the positive areas swept by the subpaths of isoperimetric profiles joining 0 to p. In particular, if the norm is strictly convex, then ∂Ω • is C 1 , and µ ≡ 0. Finally, we stress that the uniqueness of the isoperimetric profile has to be intended up to translations in R 2 .

Proof of Proposition 3.5. First of all, Proposition 3.4 tells us that γ (r,ϕ,ω) is a geodesic. We are now going to prove uniqueness assuming ω > 0, without loss of generality. As (r, ϕ, ω) ∈ R,

we can fix T > 1 such that π Ω • < ωT < 2π Ω • -δ -(ϕ).
Observe that the lower bound on ωT ensures that the area A T (γ (r,ϕ,ω) ) is bigger than half the area of Ω • (r,ϕ,ω) . Moreover, recalling the definition of δ -(ϕ), cf. (5), we have that (x T (r, ϕ, ω), y T (r, ϕ, ω)) does not lie on the flat segment of ∂Ω • (r,ϕ,ω) containing O := (0, 0). In particular, thanks to Proposition 3.7, we deduce that z T (r, ϕ, ω) > r 2 ω 2 π Ω • . On the other hand, take

ϕ ′ = ϕ + ωT -π Ω • and T ′ = 2π Ω •
ω -T and observe that, keeping in mind (6), we can explicitly compute that

x T (r, ϕ, ω), y T (r, ϕ, ω) = x T ′ (r, ϕ ′ , ω), y T ′ (r, ϕ ′ , ω) . (10) Moreover, since π Ω • < ωT < 2π Ω • -δ -(ϕ), we have that 0 < ωT ′ < π Ω • . Therefore, the area A T ′ (γ (r,ϕ ′ ,ω) ) is smaller than half the area of Ω • (r,ϕ ′ ,ω) .
In particular, we deduce that

A T ′ (γ (r,ϕ ′ ,ω) ) < r 2 ω 2 π Ω • .
Then, keeping in mind Remark 3.9 and by Proposition 3.7 and (10), we can then conclude that

z T (r, ϕ, ω) > r 2 ω 2 π Ω • > A T ′ (γ (r,ϕ ′ ,ω) ) ≥ µ x T (r, ϕ, ω
), y T (r, ϕ, ω) . Thus, by Proposition 3.8, we conclude that any geodesic connecting e and γ (r,ϕ,ω) (T ) projects to a subpath of an isoperimetric profile.

Assume by contradiction that there exist two distinct geodesics joining e and γ (r,ϕ,ω) (T ). Both geodesics must project to a subpath of an isoperimetric profile, therefore we find two distinct subpaths

Γ 1 ⊂ ∂Ω • (r,ϕ,ω) , Γ 2 ⊂ ∂Ω • (r ′ ,ϕ ′ ,ω ′ )
of isoperimetric profiles, enclosing the same signed area (thus r ω = r ′ ω ′ ), and joining O = (0, 0) and P := (x T (r, ϕ, ω), y T (r, ϕ, ω)). From Proposition 3.8, cf. Remark 3.9, there exists a translation

T v : R 2 → R 2 such that T v (x, y) = (x, y) + v, T v (∂Ω • (r,ϕ,ω) ) = ∂Ω • (r ′ ,ϕ ′ ,ω ′ ) , (11)
for some v ∈ R 2 . Now, let ℓ be the straight line through O and P and let R i be the region bounded by Γ i and ℓ, for i = 1, 2. Then, by our assumptions, R i has swept area z T (r, ϕ, ω), for i = 1, 2. Thus, as T v is a translation, the swept area of the regions R 2 and

T v (R 1 ) is the same. But T v (R 1 ) is bounded by T v (Γ 1 ) ⊂ ∂Ω • (r ′ ,ϕ ′ ,ω ′ )
and by the line T v (ℓ), hence, to not change the signed area, ℓ must be fixed by T v , meaning that v is parallel to the vector OP . Assume, without loss of generality, that v = λ OP for λ > 0. Then, we claim that

[O, P ] ⊂ ∂Ω • (r,ϕ,ω) , (12)
where [O, P ] := {sP : s ∈ [0, 1]}. Indeed, according to (11), the point

Q := (T v ) -1 (O) = -v belongs to ∂Ω • (r,ϕ,ω
) and, by construction, it is distinct from O and P . Then we find three distinct extremal points O, Q and P for the convex set Ω • (r,ϕ,ω) , belonging to ℓ. Thus the claim ( 12) is verified, by convexity.

To conclude the proof, we recall that the assumption ωT < 2π Ω •δ -(ϕ) implies that the endpoint P must not lie on the flat segment of ∂Ω • (r,ϕ,ω) containing O. This is in contradiction with (12) and concludes the proof of the first part of the statement, since T > 1 and any restriction of a geodesic is still a geodesic.

We now show the second part of the statement. If the norm • is C 1 , then C • is single valued and the whole interval C • (ϕ) are mapped to a single point. Therefore R = U .

If the norm • is strictly convex, then C • is single valued and continuous (cf. Proposition 2.16), and we can compute the limits lim ω→0

x t (r, ϕ, ω) = (r cos Ω (ϕ • ))t and lim ω→0 y t (r, ϕ, ω) = -(r sin Ω (ϕ • ))t, cf. Proposition 2.12. Moreover, since γ (r,ϕ,ω) is horizontal, then żt = 1 2 (x t ẏty t ẋt ), and we can obtain that

z t (r, ϕ, ω) = 1 2 ˆt 0 x t ẏt -y t ẋt dt = 1 2 ˆt 0 r sin Ω • (ϕ + ωt) -sin Ω • (ϕ) ω sin Ω ((ϕ + ωt) • ) + cos Ω • (ϕ + ωt) -cos Ω • (ϕ) ω cos Ω ((ϕ + ωt) • ) dt
converges to 0 as ω → 0, by the dominated convergence theorem. Finally, if the norm • is C 1 and strictly convex, then µ ≡ 0 (cf. Remark 3.9) and R = U . Therefore, according to the first part of the statement, for every p ∈ {(x, y, z) ∈ H : z = 0, (x, y) = (0, 0)}, there exists a unique geodesic of the form γ (r,ϕ,ω) for a unique (r, ϕ, ω) ∈ U , joining e and p. This implies that G 1 is a bijection onto the subset {(x, y, z) ∈ H : z = 0, (x, y) = (0, 0)}. In addition, G 1 is homeomorphism onto its image since it is continuous and proper on U .

Remark 3.10. From Definition 3.3, we can observe that

G t (r, ϕ, ω) = G 1 (tr, ϕ, tω).
In particular, when the the norm • is C 1 and strictly convex, for every t ∈ (0, 1] Proposition 3.5 guarantees that the map

G t : R >0 × R/2π Ω • Z × (-2π Ω • /t, 2π Ω • /t) \ {0} → {(x, y, z) ∈ H : z = 0, (x, y) = (0, 0)}
is a homeomorphism. Note that this identifies a sub-Finsler analogue to the so-called cotangent injectivity domain of sub-Riemannian geometry.

The Jacobian of the exponential map

In this section we study the Jacobian of the exponential map (9), in particular when it is welldefined and its properties depending on the assumption we make on the reference norm • . This object will play a fundamental role in this work, as it will allow us to formulate strategies to address the validity of the MCP(K, N ) and CD(K, N ) conditions (cf. Section 3.3).

In this first proposition we identify the set of differentiability points of the exponential map and provide the explicit expression of the Jacobian at these points. We recall that D 0 ⊂ R/2π Ω • Z is the set of differentiability points of the functions sin Ω • and cos Ω • , cf. Section 2.2. Proposition 3.11. Let H be the sub-Finsler Heisenberg group, equipped with a norm • . Given any (r, ϕ, ω) ∈ U with ϕ ∈ D 0 , for every t ∈ [0, 1] such that ϕ + ωt ∈ D 0 , the exponential map at time t

(r, ϕ, ω) -→ G t (r, ϕ, ω) is differentiable in (r, ϕ, ω) with Jacobian J t (r, ϕ, ω) = r 3 t ω 4 2 -sin Ω • (ϕ + ωt) sin Ω (ϕ • ) + cos Ω • (ϕ + ωt) cos Ω (ϕ • ) -sin Ω (ϕ + ωt) • sin Ω • (ϕ) + cos Ω (ϕ + ωt) • cos Ω • (ϕ) -ωt sin Ω (ϕ + ωt) • cos Ω (ϕ • ) -cos Ω (ϕ + ωt) • sin Ω (ϕ • ) . (13)
Proof. Note that, since ϕ, ϕ + ωt ∈ D 0 , the trigonometric functions cos Ω • and sin Ω • are differentiable at ϕ and ϕ+ωt and, in addition, the correspondence map C • is single-valued at ϕ and ϕ+ωt.

It is then possible to apply Proposition 2.12 to explicitly differentiate the quantities x t (r, ϕ, ω), y t (r, ϕ, ω) and z t (r, ϕ, ω) (cf. ( 9)). After routine computations, we end up with (13).

Remark 3.12. Since D 0 has full

L 1 -measure in R/2π Ω • Z, Fubini's theorem implies that for L 2 - a.e. (ϕ, ψ) ∈ R/2π Ω • Z × (-2π Ω • , 2π Ω • ), we have ϕ, ϕ + ψ ∈ D 0 .
In particular, given any t, the Jacobian J t (r, ϕ, ω) is defined for L 3 -almost every (r, ϕ, ω) ∈ U . In addition, if the reference norm • is strictly convex, then, according to Proposition 2.16, we know that the map C • is everywhere single valued (and then D 0 = R/2π Ω • Z) and continuous. Therefore, the map G t is C 1 , as it is everywhere differentiable and its Jacobian (13) is continuous.

We define the reduced Jacobian as the measurable function J R : R/2π

Ω • Z × (-2π Ω • , 2π Ω • ) → R, such that for every ϕ and ψ satisfying ϕ ∈ D 0 and ϕ + ψ ∈ D 0 , we have J R (ϕ, ψ) := 2 -sin Ω • (ϕ + ψ) sin Ω (ϕ • ) + cos Ω • (ϕ + ψ) cos Ω (ϕ • ) -sin Ω (ϕ + ψ) • sin Ω • (ϕ) + cos Ω (ϕ + ψ) • cos Ω • (ϕ) -ψ sin Ω (ϕ + ψ) • cos Ω (ϕ • ) -cos Ω (ϕ + ψ) • sin Ω (ϕ • ) . (14)
According to Proposition 3.11, the Jacobian in (13), when defined, can be expressed as

J t (r, ϕ, ω) = r 3 t ω 4 J R (ϕ, ωt). ( 15
)
Remark 3.13. Along the same lines of Remark 3.12, we deduce that the reduced Jacobian J R (ϕ, ψ) is defined L 2 -almost everywhere. Then, Lebesgue differentiation theorem ensures that for L 2almost every (ϕ, ψ), we have that ϕ, ϕ + ψ ∈ D 0 and (ϕ, ψ) is a Lebesgue point for J R . Applying Fubini's theorem, we deduce the existence of a L 1 -full measure set D0 ⊂ D 0 such that, for every ϕ ∈ D0 , we have that ϕ + ψ ∈ D 0 and (ϕ, ψ) is a Lebesgue point for J R , for L 1 -almost every

ψ ∈ (-2π Ω • , 2π Ω • ).
Proposition 3.14. Let H be the sub-Finsler Heisenberg group, equipped with a norm • . Given ϕ ∈ D 0 , we have that J R (ϕ, ψ) > 0 for every ψ ∈ (π

Ω • , 2π Ω • -δ -(ϕ)) such that ϕ + ψ ∈ D 0 . In particular, if the reference norm • is C 1 , then J R (ϕ, ψ) > 0 for every ϕ ∈ D 0 and every ψ ∈ (π Ω • , 2π Ω • ) such that ϕ + ψ ∈ D 0 .
Proof. First of all, according to (3), we have that

sin Ω • (ϕ + ψ) sin Ω (ϕ • ) + cos Ω • (ϕ + ψ) cos Ω (ϕ • ) ≤ 1, (16)
sin

Ω (ϕ + ψ) • sin Ω • (ϕ) + cos Ω (ϕ + ψ) • cos Ω • (ϕ) ≤ 1, (17)
and the equalities hold if and only if ϕ

+ ψ ∈ C • • C • (ϕ). We claim that -sin Ω (ϕ + ψ) • cos Ω (ϕ • ) + cos Ω (ϕ + ψ) • sin Ω (ϕ • ) ≥ 0, (18)
with the equality holding under the same condition. On the one hand, the quantity at the left-hand side can be interpreted as the scalar product of the vectors cos Ω (ϕ • ), sin Ω (ϕ • ) and

-sin Ω (ϕ + ψ) • , cos Ω (ϕ + ψ) • . (19)
On the other hand, observe that the (Euclidean) angle between the vectors cos Ω (ϕ • ), sin Ω (ϕ • ) and cos

Ω (ϕ + ψ) • , sin Ω ((ϕ + ψ) • ) (20) is in (π, 2π) for every ψ ∈ (π Ω • , 2π Ω • -δ -(ϕ))
. Therefore, inequality (18) follows by noticing that the second vector in (19) is the second vector in (20) rotated by an (Euclidean) angle of π 2 . Keeping in mind (14), the thesis follows by putting together (16), ( 17) and (18).

The second part of the statement is trivial, because, if the reference norm

• is C 1 , we have that δ -(ϕ) = 0 for every ϕ ∈ R/2π Ω • Z.
By requiring more regularity on the reference norm • , we can prove a stronger version of Proposition 3.14, taking advantage of the following lemma.

Lemma 3.15. Assume that the reference norm • is C 1 and strongly convex. Let

D + 1 ⊂ R/2π Ω • Z be the set of angles where C • is differentiable with positive derivative. Then, D + 1 ∩ I has positive L 1 -measure, for every open interval I ⊂ R/2π Ω • Z.
Proof. According to item (iii) of Proposition 2.16, the map C • is Lipschitz and therefore it is differentiable L 1 -almost everywhere and absolutely continuous. Moreover, item (i) of Proposition 2.16 guarantees that C • is strictly increasing. In particular, given any open interval I = (a, b), we deduce that

ˆb a C ′ • (ψ) dψ = C • (b) -C • (a) > 0.
Therefore the set D + 1 ∩ I must have positive L 1 -measure.

Proposition 3.16. Let H be the sub-Finsler Heisenberg group, equipped with a C 1 and strongly convex norm • . Then, the reduced Jacobian J R is everywhere non-negative and we have J R (ϕ, ψ) = 0 if and only if ψ = 0.

Proof. From the explicit formula (14), it is easy to check that if ψ = 0, then J R (ϕ, ψ) = 0. We now focus on proving that if ψ = 0, then J R (ϕ, ψ) > 0. By the periodicity of the trigonometric functions, we can assume ϕ ∈ R/2π Ω • Z and, without loss of generality, fix ψ ∈ (0, 2π Ω • ), as the case for negative ψ is completely analogous. Proposition 3.14 ensures that J R (ϕ, ψ) > 0 for every

ϕ ∈ R/2π Ω • Z and ψ ∈ (π Ω • , 2π Ω • ).
To prove positivity for the other values of ψ, we preliminary observe that, under the hypothesis of this proposition, the map C • is Lipschitz, cf. Proposition 2.16. In particular, C • is absolutely continuous and differentiable L 1 -almost everywhere and consequently, given any ϕ ∈ R/2π Ω • Z, the function ψ → J R (ϕ, ψ) is itself absolutely continuous and differentiable L 1 -almost everywhere. For every differentiability point ψ ∈ (0,

π Ω • ] of J R (ϕ, •), we have ∂ ψ J R (ϕ, ψ) = C ′ • (ϕ + ψ)K(ϕ, ψ), (21)
where

K(ϕ, ψ) := sin Ω • (ϕ + ψ) cos Ω • (ϕ) -cos Ω • (ϕ + ψ) sin Ω • (ϕ) -ψ cos Ω • (ϕ + ψ) cos Ω (ϕ • ) + sin Ω • (ϕ + ψ) sin Ω (ϕ • ) . ( 22
)
We claim that K(ϕ, ψ) > 0 for all ψ ∈ (0, π Ω • ]. In order to prove this statement, we consider the function h : R → R defined as

t → K(ϕ + ψ -t, t),
a direct computation shows that h(0) = 0. Moreover, reasoning as before, we can deduce that h is absolutely continuous and differentiable almost everywhere with derivative:

h ′ (t) = -tC ′ • (ϕ + ψ -t) (cos Ω • (ϕ + ψ) sin Ω • (ϕ + ψ -t) -sin Ω • (ϕ + ψ) cos Ω • (ϕ + ψ -t)) .
The term between parentheses can be interpreted as the inner product between the vectors

cos Ω • (ϕ + ψ), sin Ω • (ϕ + ψ) and sin Ω • (ϕ + ψ -t), -cos Ω • (ϕ + ψ -t) ,
the latter being the rotation of cos Ω • (ϕ + ψt), sin Ω • (ϕ + ψt) by an (Euclidean) angle of π 2 . Then, the scalar product is negative for every t ∈ (0, π Ω • ). In particular, recalling Lemma 3.15 and that h is absolutely continuous, we conclude that

K(ϕ, ψ) = h(ψ) = ˆψ 0 h ′ (s) ds > 0,
proving our claim. Analogously, equation (21) combined with Lemma 3.15 allows us to deduce that J R (ϕ, ψ) > 0 for every ϕ ∈ R/2π Ω • Z and ψ ∈ (0, π Ω • ], concluding the proof.

The MCP condition in the Heisenberg group

In this section we discuss several results that will help us in addressing the validity of the MCP(K, N ) condition in the sub-Finsler Heisenberg groups. In these results the Jacobian of the exponential map studied in the previous section plays a fundamental role, as it describes the infinitesimal volume contraction along geodesics. First of all, we show that, since the Heisenberg group admits a one-parameter family of dilations, it is sufficient to study the validity of the MCP(K, N ) (and the CD(K, N ) condition) with curvature parameter K = 0 and having L 3 as reference measure.

Proposition 3.17. Let H be the sub-Finsler Heisenberg group, equipped with a norm • , and with a smooth measure m. Assume that the metric measure space (H, d, m) satisfies the MCP(K, N ) (resp. CD(K, N )) condition, for some K ∈ R and N ∈ (1, ∞). Then, the metric measure (H, d, L 3 ) satisfies the MCP(0, N ) (resp. CD(0, N )) condition.

Proof. Let m : H → R >0 be the (smooth) density of the measure m with respect to the Lebesgue measure L 3 . Since the Heisenberg group admits a one-parameter family of dilations [START_REF] Le | A metric characterization of Carnot groups[END_REF], it holds that

H, n • d, n 4 • m, e pmGH ----→ (H, d, m(e) • L 3 , e) as n → ∞.
Moreover, by the scaling property of the MCP condition, we have that H, n • d, n 4 • m is a MCP(K/n 2 , N ) space. Then, the pmGH-stability of the MCP condition ensures that (H, d, m(e) • L 3 ) is a MCP(0, N ) space. Using once again the scaling property, we conclude that (H, d, L 3 ) is a MCP(0, N ) spaces as well. The same argument, proves the result for the CD condition.

Second of all, we characterize the measure contraction property MCP(K, N ) in terms of the reduced Jacobian of the exponential map.

Proposition 3.18. Let H be the sub-Finsler Heisenberg group, equipped with a norm • , and with the Lebesgue measure L 3 . If the metric measure space (H, d, L 3 ) satisfies the measure contraction property MCP(0, N ), then

|J R (ϕ, ωt)| ≥ t N -1 |J R (ϕ, ω)|,
for every (r, ϕ, ω) ∈ R and t ∈ [0, 1] such that ϕ, ϕ + ωt, ϕ + ω ∈ D 0 and (ϕ, ωt), (ϕ, ω) are Lebesgue points for J R .

Proof. Fix any (r, ϕ, ω) ∈ R and t as in the hypothesis and assume that MCP(0, N ) holds. For every ε > 0 sufficiently small, we consider the ball B ε (r, ϕ, ω) of radius ε (with respect to the Euclidean distance) centred at the point (r, ϕ, ω). Observe that Proposition 3.5 ensures that for every x ∈ G 1 B ε (r, ϕ, ω) there exists a unique geodesic connecting e to x and it is of the form γ (r ′ ,ϕ ′ ,ω ′ ) , for some (r ′ , ϕ ′ , ω ′ ) ∈ B ε (r, ϕ, ω). Therefore, the characterization (2) yields that for every ε (sufficiently small) we have that

L 3 G t B ε (r, ϕ, ω) ≥ t N L 3 G 1 B ε (r, ϕ, ω) .
Then, thanks to our assumptions on the parameters and keeping in mind (15), we deduce that

|J t (r, ϕ, ω)| = lim ε→0 + 1 L 3 (B ε (r, ϕ, ω)) ˆBε(r,ϕ,ω) |J t (r ′ , ϕ ′ , ω ′ )| dr ′ dϕ ′ dω ′ = lim ε→0 + L 3 G t B ε (r, ϕ, ω) L 3 (B ε (r, ϕ, ω)) ≥ lim ε→0 + t N L 3 G 1 B ε (r, ϕ, ω) L 3 (B ε (r, ϕ, ω)) = t N lim ε→0 + 1 L 3 (B ε (r, ϕ, ω)) ˆBε(r,ϕ,ω) |J 1 (r ′ , ϕ ′ , ω ′ )| dr ′ dϕ ′ dω ′ = t N |J 1 (r, ϕ, ω)|,
where in the second and third equality we were able to use the area formula because G t is locally Lipschitz for every t. The thesis immediately follows from (15).

Combining Proposition 3.17 and Proposition 3.18, we immediately deduce the following corollary, that provides an effective strategy to disprove the measure contraction property.

Corollary 3.19. Let H be the sub-Finsler Heisenberg group, equipped with a norm • , and with a smooth measure m. Suppose there exist (r, ϕ, ω) ∈ R and t ∈ [0, 1], such that ϕ, ϕ + ωt, ϕ + ω ∈ D 0 , (ϕ, ωt), (ϕ, ω) are Lebesgue points for J R and

|J R (ϕ, ωt)| < t N -1 |J R (ϕ, ω)|.
Then, the metric measure space (H, d, m) does not satisfy the measure contraction property MCP(K, N ), for every K ∈ R and N ∈ (1, ∞).

In the following statement we equivalently characterize the validity of the MCP(0, N ) condition, when the reference norm • is C 1 and strictly convex. In the sequel, we denote by U the projection of U in the variables (ϕ, ω), namely

U := R/2π Ω • Z × (-2π Ω • , 2π Ω • ) \ {0}.
Proposition 3.20. Let H be the sub-Finsler Heisenberg group, equipped with a C 1 and strictly convex norm • , and with the Lebesgue measure L 3 . Then, the metric measure space (H, d, L 3 ) satisfies the MCP(0, N ) if and only if

|J R (ϕ, ωt)| ≥ t N -1 |J R (ϕ, ω)|, (23)
for every (ϕ, ω) ∈ U and every t ∈ [0, 1].

Remark 3.21. The left-translations (8) are isometries and L 3 is a left-invariant measure. As a consequence, in order to test the validity of the measure contraction property MCP(0, N ), it is sufficient to prove the condition (cf. Definition 2.4 and Remark 2.5) for x = e.

Proof of Proposition 3.20. The "only if" part is a consequence of Proposition 3.18 and Proposition 3.16. In fact, in the case we are considering, we have that

D 0 = R/2π Ω • Z, R = U and the reduced Jacobian J R (•, •
) is defined and continuous on U .

For the "if" part, we assume that (23) holds. Given any Borel set A ⊂ H, we observe that A ⊂ {(x, y, z) ∈ H : z = 0, (x, y) = (0, 0)} up to a L 3 -null set. In particular, according to the last part of Proposition 3.5, there exists a Borel set B ⊂ U such that A = G 1 (B) up to a L 3 -null set and G 1 is an homeomorphism between B and G 1 (B). Moreover, Remark 3.10 guarantees that G t is injective for every t ∈ (0, 1]. Then, using the area formula and (23), we deduce that

L 3 (M t (e, A)) = L 3 (G t (B)) = ˆB |J t (r, ϕ, ω)| dr dϕ dω = ˆB r 3 t ω 4 |J R (ϕ, tω)| dr dϕ dω ≥ t N ˆB r 3 ω 4 |J R (ϕ, ω)| dr dϕ dω = t N ˆB |J 1 (r, ϕ, ω)| dr dϕ dω = t N L 3 (G 1 (B)) = t N L 3 (A).
Finally, we note that this inequality is sufficient to prove the MCP(0, N ) condition, according to Remarks 2.5 and 3.21.

We provide an alternative version of Proposition 3.20, under the further assumption that the norm • is strongly convex. In this case, Proposition 3.20, combined with Proposition 3.16, guarantees that the MCP(0, N ) condition holds if and only if

J R (ϕ, ωt) ≥ t N -1 J R (ϕ, ω), (24)
for every (ϕ, ω) ∈ U and every t ∈ [0, 1]. Moreover, when • is C 1 and strongly convex, Proposition 2.16 ensures the map C • is Lipschitz and differentiable L 1 -almost everywhere. We call D 1 ⊂ R/2π Ω • Z the (full-measure) set of differentiability points.

Corollary 3.22. Let H be the sub-Finsler Heisenberg group, equipped with a C 1 and strongly convex norm • , and with the Lebesgue measure L 3 . Then, the metric measure space (H, d, L 3 ) satisfies the MCP(0, N ) if and only if

N (ϕ, ω) := 1 + ω∂ ω J R (ϕ, ω) J R (ϕ, ω) ≤ N, (25)
for all (ϕ, ω) ∈ U with ϕ + ω ∈ D 1 .

Proof. It is sufficient to prove that (25) is equivalent to (24). Assume the latter holds and let (ϕ, ω) ∈ U such that ϕ + ω ∈ D 1 . Then, J R (ϕ, •) is differentiable at ω and we obtain that

∂ ω J R (ϕ, ω) J R (ϕ, ω) = d dω log(J R (ϕ, ω)) = lim z→ω log(J R (ϕ, z)) -log(J R (ϕ, ω)) z -ω = 1 ω lim t→1 - log(J R (ϕ, ωt)) -log(J R (ϕ, ω)) t -1 ≤ 1 ω lim t→1 - (N -1) log t t -1 = N -1 ω ,
where the inequality follows from (24).

For the other implication, observe that, by Proposition 2.16, C • is Lipschitz and therefore map J R (ϕ, •) is Lipschitz and thus absolutely continuous. Then, (25) implies that

ˆω ωt d dz log(J R (ϕ, z)) dz ≤ (N -1) ˆω ωt d dz log z dz,
which simplifies to (24).

We conclude the section with a useful criterion that ensures the validity of the MCP(0, N ) condition for some N ∈ (1, ∞).

Proposition 3.23. Let H be the sub-Finsler Heisenberg group, equipped with a C 1 and strongly convex norm • , and with the Lebesgue measure L 3 . Then, the metric measure space (H, d, L 3 ) satisfies the MCP(0, N ) for some N ∈ (1, ∞) if and only if for any

ϕ ∞ ∈ R/2π Ω • Z, lim sup (ϕ,ω)→(ϕ∞,0) N (ϕ, ω) < +∞, ( 26
)
where the lim sup is taken over all (ϕ, ω) ∈ U such that ϕ + ω ∈ D 1 .

Proof. The "only if" part is follows immediately from Corollary 3.22, as (25) implies (26). Let us investigate the "if" part of the statement. For k ∈ N, denote by U k the set

U k := {(ϕ, ω) ∈ U | ϕ + ω ∈ D 1 and N (ϕ, ω) ≥ k}.
By Corollary 3.22, satisfying the MCP(0, N ) condition for some N ∈ (1, ∞) is equivalent to the existence of a k ∈ N such that U k = ∅. Assume by contradiction that for all k ∈ N, the set U k is not empty. Then, there exists a sequence {ϕ k , ω k } k∈N ⊂ U such that lim k→+∞ N (ϕ k , ω k ) = +∞ and ϕ k + ω k ∈ D 1 for all k ∈ N. Up to extracting a converging subsequence, we can assume that (ϕ k , ω k ) converges to (ϕ ∞ , ω ∞ ). Firstly, we claim that ω ∞ = 0, ±2π Ω • . Indeed, if this were not the case, keeping in mind (21), ( 22) and that C • is Lipschitz, we would have that

N (ϕ k , ω k ) = 1 + C ′ • (ϕ k + ω k ) ω k K(ϕ k , ω k ) J R (ϕ k , ω k ) ≤ 1 + Lip(C • ) ω k K(ϕ k , ω k ) J R (ϕ k , ω k ) → 1 + Lip(C • ) ω ∞ K(ϕ ∞ , ω ∞ ) J R (ϕ ∞ , ω ∞ ) < ∞ as k → +∞,
where we used continuity of ωK(ϕ, ω)/J R (ϕ, ω) and that the last term is finite because Proposition 3.16 guarantees that J R (ϕ ∞ , ω ∞ ) > 0. Moreover, we can exclude that ω ∞ = ±2π Ω • because ωK(ϕ, ω) < 0 near ω = ±2π Ω • , while C ′ • (ϕ + ω) ≥ 0 and J R (ϕ, ω) > 0 by Proposition 3.16. Finally, ω ∞ = 0 is excluded by the assumption (26) and we obtain the desire contradiction.

O ∂Ω • (x, y 1 ) = Q ψ 1 (x, y 2 ) = Q φ (x, y 0 ) = Q ψ 0 I Figure 3: The flat part of ∂Ω • .
4 Failure of the measure contraction property in the sub-Finsler Heisenberg group

In this section we study the validity of the measure contraction property MCP(K, N ) in the sub-Finsler Heisenberg group H, equipped with the norm • . In particular, we are going to prove Theorem 1.1, as a result of the combination of Theorem 4.1 and Theorem 4.5, and Theorem 1.2.

Failure for non-C 1 norms

In this first subsection we prove Theorem 1.1 for every non-C 1 reference norm • . The proof we are going to present follows the same strategy developed in [MR23b, Thm. 5.26], properly adapted taking advantage of Proposition 3.5, in order to include the case of non-strictly convex norms. As in [MR23b, Thm. 5.26], the main idea is to exploit a branching behaviour of geodesics which is caused by the singularities of the reference norm • .

Theorem 4.1. Let H be the sub-Finsler Heisenberg group, equipped with a norm • which is not C 1 , and let m be a smooth measure on H. Then, the metric measure space (H, d, m) does not satisfy the measure contraction property MCP(K, N ) for every K ∈ R and N ∈ (1, ∞).

Proof. According to Proposition 2.14, since • is not C 1 , its dual norm • * is not strictly convex.

In particular, there exists a straight segment contained in the sphere S

• * 1 (0) = ∂Ω • . Since the distribution generating the Heisenberg group is invariant under rotations around the z-axis, we can assume without losing generality that this segment is vertical in R 2 ∩ {x > 0}, i.e. there exists x ∈ R >0 and an interval

I := [y 0 , y 1 ] ⊂ R such that {x} × I ⊂ ∂Ω • .
Moreover, we can take the interval I to be maximal, namely for every y ∈ I we have (x, y) ∈ Ω • . Let ψ 0 , ψ 1 ∈ R/2π Ω • Z be such that Q ψ 0 = (x, y 0 ) and Q ψ 1 = (x, y 1 ) (see Figure 3). Observe that, for every y ∈ I, it holds that (x, y)

= Q ψ 0 +(y-y 0 )x , (27)
as a consequence, we deduce that cos Ω • (ψ 0 + (yy 0 )x) = x and sin Ω • (ψ 0 + (yy 0 )x) = y, for y ∈ I.

Consider φ := 1 2 (ψ 0 + ψ 1 ) and y 2 = 1 2 (y 0 + y 1 ), so that (x, y 2 ) = Q φ, according to (27). Fix r > 0 and ω > π Ω • such that (r, φ, ω) ∈ R and take a neighborhood A ⊂ R of (r, φ, ω), such that ω > π Ω • for every (r, ϕ, ω) ∈ A . Proposition 3.5 guarantees that, for every (r, ϕ, ω) ∈ A , the curve γ (r,ϕ,ω) is the unique geodesic connecting e and G 1 (r, ϕ, ω). Therefore, we deduce that

M t {e}, G 1 (A ) = G t (A ), for every t ∈ [0, 1].
As observed in Remark 3.12, we have ϕ, ϕ + tω, ϕ + ω ∈ D 0 for L 3 -a.e. (r, ϕ, ω) ∈ A . Moreover, at those points, Proposition 3.14 ensures that

J 1 (r, ϕ, ω) = r 3 ω 4 J R (ϕ, ω) > 0. ( 29
)
Therefore, since G 1 (•) is locally Lipschitz on A , by the area formula and (29), we deduce that

m G 1 (A ) > 0. ( 30
)
We can now disprove the measure contraction property MCP(K, N ), taking as marginals

µ 0 := δ e and µ 1 := 1 L 3 (G 1 (A )) L 3 | G 1 (A ) .
According to Remark 2.5, it is enough to contradict (2) with A ′ = A = G 1 (A ). In particular, we are going to find t 0 ∈ (0, 1) such that

M t ({e}, A) = G t (A ) ⊂ {y = 0, z = 0}, ∀ t < t 0 . (31)
To this aim, fix any (r, ϕ, ω) ∈ A and note that, for every t < ψ 1 -ϕ ω , (28) implies that

cos Ω • (ϕ + ωt) = x and sin Ω • (ϕ + ωt) = sin Ω • (ϕ) + ωt x .
From these relations and (9), it follows immediately that y t (r, ϕ, ω) = 0 and z t (r, ϕ, ω) = 0, for every s < ψ 1 -ϕ ω . Then, according to our choice of A , (31) holds with t 0 = ψ 1 -ϕ π Ω • .

Remark 4.2. Observe that the previous theorem generalizes [MR23b, Thm. 5.26]. While the strategy of its proof is similar, we highlight that now we are able to prove (30) using Proposition 3.14 and working in a neighborhood A where ω > π Ω • . This is in contrast with the geometric construction of [MR23b, Thm. 5.26] which was local around the flat part of ∂Ω • .

Remark 4.3. As already observed in [MR23b, Rmk. 5.27], the construction presented in the last proof highlights the existence of a family of branching geodesics, originating from the presence of a flat part in ∂Ω • . In particular, when H is equipped with a singular norm, geodesics can branch, although they are unique (in the sense of Proposition 3.5).

Failure for C 1 and non-strictly convex norms

In this subsection we complete another step for the proof of Theorem 1.1, considering the sub-Finsler Heisenberg group H, equipped with a non-strictly convex C 1 norm • . The main idea behind our strategy is to exploit the discontinuities of the generalized trigonometric functions, caused by the "flat parts" of ∂Ω (cf. Lemma 4.4). According to Proposition 3.11, these will result in discontinuities of the Jacobian of the exponential map, which, if properly utilized, will allow us to show the failure of the MCP(K, N ) condition.

In particular, in this case there exists a straight segment L contained in the sphere S

• 1 (0) = ∂Ω. As done in the proof of Theorem 4.1, we can assume without losing generality that this segment is vertical in R 2 ∩ {x > 0}, i.e. there exists x ∈ R >0 and a maximal interval

I := [y 0 , y 1 ] ⊂ R such that L = {x} × I ⊂ ∂Ω.
Lemma 4.4. We have that

lim ψ↑0 ψ∈D 0 cos Ω (ψ • ) = lim ψ↓0 ψ∈D 0 cos Ω (ψ • ) = x and lim ψ↑0 ψ∈D 0 sin Ω (ψ • ) = y 0 < y 1 = lim ψ↓0 ψ∈D 0 sin Ω (ψ • ).
Proof. Let v ∈ ∂Ω • be the dual vector of every vector in L, i.e. v = d (x,y) • , for every y ∈ I.

Notice that v is an horizontal vector in R 2 , therefore v = Q 0 . In particular, setting θ 0 and θ 1 to be such that (x, y 0 ) = P θ 0 and (x, y 1 ) = P θ 1 , we have

C • (θ) = 0 if and only if θ ∈ [θ 0 , θ 1 ]. ( 32
)
Fix ε > 0 and let θ0 < θ 0 and θ1 > θ 1 such that

cos Ω (θ) ∈ (x -ε, x], for every θ ∈ [ θ0 , θ 0 ] ∪ [θ 1 , θ1 ],
sin Ω (θ) ∈ (y 0ε, y 0 ], for every θ ∈ [ θ0 , θ 0 ],

sin

Ω (θ) ∈ [y 1 , y 1 + ε), for every θ ∈ [θ 1 , θ1 ].
Call ψ 0 := C • ( θ0 ) < 0 and ψ 1 := C • ( θ1 ) > 0, cf. (32). Now, consider ψ ∈ (ψ 0 , 0) ∩ D 0 (so that C • is a single-valued map at ψ) and note that C

• (ψ) = ψ • ∈ [ θ0 , θ 0 ] and therefore it holds that cos Ω (ψ • ) ∈ (x -ε, x], sin Ω (ψ • ) ∈ (y 0 -ε, y 0 ]. (33)
With an analogous argument, we deduce that for every ψ ∈ (0, ψ 1 )∩ D 0 , we have that ψ

• ∈ [θ 1 , θ1 ] and cos Ω (ψ • ) ∈ (x -ε, x], sin Ω (ψ • ) ∈ [y 1 , y 1 + ε). (34)
By the arbitrariness of ε > 0, the combination of (33) and (34) yields the thesis.

Theorem 4.5. Let H be the sub-Finsler Heisenberg group, equipped with a norm • which is C 1 and not strictly convex, and let m be a smooth measure on H. Then, the metric measure space (H, d, m) does not satisfy the measure contraction property MCP(K, N ) for every K ∈ R and N ∈ (1, ∞).

Before going through the proof, we recall from section 3.2 that there is a L 1 -full measure set D0 ⊂ D 0 such that, for every ϕ ∈ D0 , we have that ϕ + ψ ∈ D 0 and (ϕ, ψ) is a Lebesgue point for J R , for L 1 -almost every ψ ∈ R.

Proof of Theorem 4.5. According to Lemma 4.4, the following limits exist

y 0 = lim ψ↑0 ψ∈D 0 sin Ω (ψ • ) and y 1 = lim ψ↓0 ψ∈D 0 sin Ω (ψ • ),
and we know that δ := y 1y 0 > 0. Observe that, by the symmetry of the norm, cos

Ω • (π Ω • ) = -cos Ω • (0) < 0 and sin Ω • (π Ω • ) = sin Ω • (0) = 0.
Let ρ > 0 be sufficiently small, then, by continuity of the trigonometric functions, there exists ε > 0 such that, for every ϕ

∈ [π Ω • -ε, π Ω • ],
sin

Ω • (ϕ) ∈ 0, ρ M and -cos Ω • (ϕ) ∈ [cos Ω • (0) -ρ, cos Ω • (0) + ρ], (35)
where M := max θ sin Ω (θ) is positive and finite. Thus, by the Pythagorean identity (3) and the first relation in (35), we deduce that cos Ω (ϕ

• ) cos Ω • (ϕ) ∈ [1 -ρ, 1 + ρ], ∀ ϕ ∈ [π Ω • -ε, π Ω • ] ∩ D 0 .
Combining this with the second relation in (35), we deduce that

-cos Ω (ϕ • ) ∈ 1 -ρ cos Ω • (0) + ρ , 1 + ρ cos Ω • (0) -ρ , ∀ ϕ ∈ [π Ω • -ε, π Ω • ] ∩ D 0 .
In addition, up to restricting ε > 0, by continuity, we may also assume that

2 π Ω • sin Ω • (ϕ) < 1 -ρ cos Ω • (0) + ρ , ∀ ϕ ∈ [π Ω • -ε, π Ω • ] ∩ D 0 .
In conclusion, we can find ε > 0 such that

-cos Ω (ϕ • ) > 2 π Ω • sin Ω • (ϕ) for every ϕ ∈ [π Ω • -ε, π Ω • ] ∩ D 0 . ( 36
) Fix φ ∈ [π Ω • -ε, π Ω • )∩ D0 , call k(ψ) := -sin Ω • ( φ)-ψ cos Ω ( φ• ) and observe that (36) ensures that k(ψ) > -sin Ω • ( φ) - π Ω • 2 cos Ω ( φ• ) > 0 for every ψ ≥ π Ω • 2 . ( 37
)
Note that the function ψ → J R ( φ, ψ) is not continuous at 2π Ω •φ. In fact, looking at the explicit expression ( 14) of the reduced Jacobian, thanks to Lemma 4.4, we have that every term of J R (ϕ, ψ) is continuous (in ψ, at 2π Ω •φ) except for

-sin Ω ( φ + ψ) • sin Ω • ( φ) -ψ sin Ω ( φ + ψ) • cos Ω ( φ• ) = k(ψ) sin Ω (ϕ + ψ) • .
As a consequence, calling

J 1 := lim ψ↑(2π Ω • -φ) φ+ψ∈D 0 J R ( φ, ψ) and J 2 := lim ψ↓(2π Ω • -φ) φ+ψ∈D 0 J R ( φ, ψ), (38)
we deduce that

J 2 > J 2 -k(2π Ω • -φ) • δ = J 1 > 0
where the first inequality is a consequence (37), while the second one follows from Proposition 3.14. Recall that, since we chose φ ∈ D0 , we have that, for L 1 -almost every ψ ∈ R, φ + ψ ∈ D 0 and (ϕ, ψ) is a Lebesgue point for J R . Now, we consider the quantity

t := J 2 -2 3 k(2π Ω • -φ) J 2 -1 3 k(2π Ω • -φ) < 1.
Keeping in mind (38), for any N ∈ (1, ∞), we can find

ψ 1 < 2π Ω • -φ and ψ 2 > 2π Ω • -φ such that φ + ψ 1 , φ + ψ 2 ∈ D 0 , ( φ, ψ 1 ) and ( φ, ψ 2 ) are Lebesgue points for J R , ψ 1 ψ 2 =: t > t 1 N-1 , J R ( φ, ψ 1 ) < J 1 + 1 3 k(2π Ω • -φ) = J 2 - 2 3 k(2π Ω • -φ) and J R ( φ, ψ 2 ) > J 2 - 1 3 k(2π Ω • -φ).
Now, taking ω = ψ 2 , we conclude that

J R ( φ, ωt) = J R ( φ, ψ 1 ) < J 2 - 2 3 k(2π Ω • -φ) = t J 2 - 1 3 k(2π Ω • -φ) < t • J R ( φ, ψ 2 ) < t N -1 • J R ( φ, ω).
Then, we can apply Corollary 3.19 and deduce the thesis, thanks to the arbitrariness of N .

Failure for C 1 and strictly but not strongly convex norms

In this subsection, we consider the sub-Finsler Heisenberg group H, equipped with a norm • which is C 1 and strictly convex, but not strongly convex and prove the failure of the MCP(K, N ) condition in these structures. In this way, we complete the proof of Theorem 1.1. The strategy used in this section is similar to the one developed to prove Theorem 4.5. But, instead of looking at discontinuities of the Jacobian, which is continuous in this case, we exploit the existence of suitable parameters at which the Jacobian has big variations. This case includes the Heisenberg group equipped with the sub-Finsler ℓ p -norm when p ∈ (2, +∞), and recovers the result in [BT23, Thm. A.2].

Theorem 4.6. Let H be the sub-Finsler Heisenberg group, equipped with a C 1 and strictly but not strongly convex norm • , and with a smooth measure m. Then, the metric measure space (H, d, m) does not satisfy the MCP(K, N ) for every K ∈ R and N ∈ (1, ∞).

Before going to the proof, we observe that the characterization (23) of Proposition 3.20 is equiv-

alent to log |J R (ϕ, ωt)| -log |J R (ϕ, ω)| ωt -ω ≤ (N -1) log t ωt -ω ,
for every (ϕ, ω) ∈ U and t ∈ [0, 1]. In particular, if (H, d, L 3 ) satisfies the MCP(0, N ) for some

N ∈ (1, ∞), then lim sup (ω 1 ,ω 2 )→(ω,ω) ω 1 =ω 2 log |J R (ϕ, ω 1 )| -log |J R (ϕ, ω 2 )| ω 1 -ω 2 ≤ (N -1) lim sup (ω 1 ,ω 2 )→(ω,ω) ω 1 =ω 2 log |ω 1 | -log |ω 2 | ω 1 -ω 2 = N -1 ω < +∞, ( 39
)
for every (ϕ, ω) ∈ U . The last equality is justified since lim

(x,y)→(a,a) x =y f (x) -f (y) x -y = f ′ (a),
whenever f is C 1 in a neighbourhood of a (by the mean value theorem). In the proof, we will also use the property lim sup

(x,y)→(a,a) x =y f (g(x)) -f (g(y)) g(x) -g(y) = lim sup (x,y)→(g(a),g(a)) x =y f (x) -f (y) x -y ( 40
)
whenever g is strictly monotone and continuous.

Proof of Theorem 4.6. According to Proposition 2.16, we know that the angle correspondence C • is strictly increasing and continuous but not Lipschitz. Then, we can fix ψ ∈ R/2π

C • (ψ n ) -C • (η) ψ n -η = ∞
, and thus lim sup

(x,y)→(ψ,ψ) x =y C • (x) -C • (y) x -y = +∞.
We set ω := π Ω • and ϕ := ψω and we shall see that J R (ϕ, •) fails to be Lipschitz at ω. For every ω 1 , ω 2 near ω, we can explicitly compute that

J R (ϕ, ω 1 ) -J R (ϕ, ω 2 ) ω 1 -ω 2 = - cos Ω • (ϕ + ω 1 ) -cos Ω • (ϕ + ω 2 ) ω 1 -ω 2 cos Ω (ϕ • ) - sin Ω • (ϕ + ω 1 ) -sin Ω • (ϕ + ω 2 ) ω 1 -ω 2 sin Ω (ϕ • ) -cos Ω (ϕ • ) sin Ω ((ϕ + ω 1 ) • ) + sin Ω (ϕ • ) cos Ω ((ϕ + ω 1 ) • ) + C • (ϕ + ω 1 ) -C • (ϕ + ω 2 ) ω 1 -ω 2 × F (ϕ, ω 1 , ω 2 ) C • (ϕ + ω 1 ) -C • (ϕ + ω 2 )
,

where

F (ϕ, ω 1 , ω 2 ) := sin Ω (C • (ϕ + ω 1 )) -sin Ω (C • (ϕ + ω 2 )) (-sin Ω • (ϕ) -ω 2 cos Ω (ϕ • )) + cos Ω (C • (ϕ + ω 1 )) -cos Ω (C • (ϕ + ω 2 )) (-cos Ω • (ϕ) + ω 2 sin Ω (ϕ • )).
Note that the continuity of C • yields that lim

(ω 1 ,ω 2 )→(ω,ω) ω 1 =ω 2 sin Ω ((ϕ + ω 1 ) • ) = sin Ω ((ϕ + ω) • ) and lim (ω 1 ,ω 2 )→(ω,ω) ω 1 =ω 2 cos Ω ((ϕ + ω 2 ) • ) = cos Ω ((ϕ + ω) • ).
Moreover, since the generalised trigonometric functions cos Ω • and sin Ω • are C 1 , we also have that lim

(ω 1 ,ω 2 )→(ω,ω) ω 1 =ω 2 cos Ω • (ϕ + ω 1 ) -cos Ω • (ϕ + ω 2 ) ω 1 -ω 2 = -sin Ω ((ϕ + ω) • ),
and, similarly, lim

(ω 1 ,ω 2 )→(ω,ω) ω 1 =ω 2 sin Ω • (ϕ + ω 1 ) -sin Ω • (ϕ + ω 2 ) ω 1 -ω 2 = cos Ω ((ϕ + ω) • ).
Finally, since C • is strictly increasing and continuous, we find that lim

(ω 1 ,ω 2 )→(ω,ω) ω 1 =ω 2 sin Ω (C • (ϕ + ω 1 )) -sin Ω (C • (ϕ + ω 2 )) C • (ϕ + ω 1 ) -C • (ϕ + ω 2 ) = lim (ϕ 1 ,ϕ 2 )→((ϕ+ω)•,(ϕ+ω)•) ϕ 1 =ϕ 2 sin Ω (ϕ 1 ) -sin Ω (ϕ 2 ) ϕ 1 -ϕ 2 = cos Ω • (ϕ + ω)
where the first inequality follows from (40), while the second is a consequence of the differentiability of sin Ω , cf. Proposition 2.12. In the same way, we also have that lim

(ω 1 ,ω 2 )→(ω,ω) ω 1 =ω 2 cos Ω (C • (ϕ + ω 1 )) -cos Ω (C • (ϕ + ω 2 )) C • (ϕ + ω 1 ) -C • (ϕ + ω 2 ) = -sin Ω • (ϕ + ω).
As a consequence, keeping in mind (6), we obtain that lim

(ω 1 ,ω 2 )→(ω,ω) ω 1 =ω 2 F (ϕ, ω 1 , ω 2 ) C • (ϕ + ω 1 ) -C • (ϕ + ω 2 ) = cos Ω • (ϕ + ω)(-sin Ω • (ϕ) -ω cos Ω (ϕ • )) -sin Ω • (ϕ + ω)(-cos Ω • (ϕ) + ω sin Ω (ϕ • )) = ω > 0.
Putting everything together, since C • is not Lipschitz at ψ = ϕ + ω, we deduce that J R (ϕ, •) fails to be Lipschitz at ω. Furthermore, the logarithm function log is locally bi-Lipschitz and the value of J R (ϕ, •) stays in a compact subset of (0, +∞) near ω, therefore also log(J R (ϕ, •)) fails to be Lipschitz at ω. We conclude that the Heisenberg group is not MCP(0, N ) for any N ∈ (0, ∞), as (39) is not satisfied. The thesis then follows from Proposition 3.17.

Measure contraction property for C 1 and strongly convex norms

In this section, we study the measure contraction property for the sub-Finsler Heisenberg group equipped with a norm • that is C 1 and strongly convex. Firstly, we provide further necessary (but not sufficient) conditions for the MCP to hold in this case. Secondly, we identify sufficient (but not necessary) conditions for the validity of MCP, providing estimates for the curvature exponent (cf. Definition 2.6). In particular, we show that MCP holds under C 1,1 regularity of the reference norm • . Thirdly, we investigate possible necessary and sufficient conditions for MCP through several meaningful examples. This study relies on the behavior of (the derivative of) the angle correspondence C • . We summarize the results of this section as follows.

(i) If C ′

• has discontinuous zero points or non-negligible zero points, then (H, d, L 3 ) does not satisfy MCP (Theorems 5.2 and 5.3). This happens, for example, when C ′ • (ϕ) = 1 Z c (ϕ) near one of its zero points, where Z is a fat Cantor set.

(ii) If C ′

• has infinite order at one of its zero points, then (H, d, L 3 ) does not satisfy MCP (Theorem 5.4). This happens, for example, when

C ′ • (ϕ) = e -1/|ϕ| near ϕ = 0. (iii) If C ′
• is uniformly asymptotic to a fractional polynomial near its zero points, then (H, d, L 3 ) satisfies MCP(0, N ) for some N ∈ (1, +∞) (Theorem 5.9 and 5.10). This happens, for example, for the sub-Finsler Heisenberg group equipped with the ℓ p -norm with p ∈ (1, 2).

(iv) If C ′
• is not a fractional polynomial but is "monotone" near its zero points, then there are examples of sub-Finsler structures on the Heisenberg group that satisfy the MCP(0, N ) for any N ∈ (1, +∞) , see Example 5.16.

(v) If C ′
• "oscillates with large variation" near one of its zero points, then there are examples of sub-Finsler structures on the Heisenberg group that do not satisfy the MCP(0, N ) for any N ∈ (1, +∞) , see Example 5.17.

Zero points of C ′

• and non MCP Under the assumptions of this section, Proposition 2.16 ensures that the angle C • strictly increasing and Lipschitz continuous (thus differentiable almost everywhere). Let Z = D 1 \D + 1 ⊂ R/2π Ω • Z be the set of zero points of C ′

• . Under the assumption on the norm • to be C 1 and strongly convex, we do not have information on the size of Z and the behavior of C ′ • near its zero points. In this section, we investigate how the zero points in Z affect the validity (or failure) of the measure contraction property. We start with a lemma which gives a lower bound of the crucial ratio ω∂ ω J R /J R solely using C • .

Lemma 5.1. Let H be the sub-Finsler Heisenberg group, equipped with a C 1 and strongly convex norm • . For a fixed angle ϕ ⋆ ∈ R/2π Ω • Z, we have lim sup

(ϕ,ω)→(ϕ ⋆ ,0) ω∂ ω J R (ϕ, ω) J R (ϕ, ω) ≥ lim sup (ϕ,ω)→(ϕ ⋆ ,0) ωC ′ • (ϕ + ω) C • (ϕ + ω) -C • (ϕ) ,
where the lim sup is taken over all (ϕ, ω) ∈ U such that ϕ + ω ∈ D 1 . In particular, if there is

ϕ ⋆ ∈ R/2π Ω • Z such that lim sup (ϕ,ω)→(ϕ ⋆ ,0) ωC ′ • (ϕ + ω) C • (ϕ + ω) -C • (ϕ) = +∞,
then the metric measure space (H, d, L 3 ) does not satisfy the MCP(0, N ) for any N ∈ (1, ∞).

Proof. By using the function K(ϕ, ω) defined in (22), we can write

ω∂ ω J R (ϕ, ω) J R (ϕ, ω) = ωC ′ • (ϕ + ω) C • (ϕ + ω) -C • (ϕ) (C • (ϕ + ω) -C • (ϕ))K(ϕ, ω) J R (ϕ, ω) . We claim that (C • (ϕ + ω) -C • (ϕ))K(ϕ, ω) J R (ϕ, ω) ≥ 1, (41)
for sufficiently small ω ∈ D 1 . We prove (41) for ω > 0, as the proof for ω < 0 is completely analogous. At every differentiable point ϕ + ω ∈ D 1 , we have

∂ ω (C • (ϕ + ω) -C • (ϕ))K(ϕ, ω) -J R (ϕ, ω) = (C • (ϕ + ω) -C • (ϕ))∂ ω K(ϕ, ω) = (C • (ϕ + ω) -C • (ϕ)) sin Ω ((ϕ + ω) • ) sin Ω • (ϕ) + ω cos Ω (ϕ • ) + cos Ω ((ϕ + ω) • ) cos Ω • (ϕ) -ω sin Ω (ϕ • ) -cos Ω • (ϕ + ω) cos Ω (ϕ • ) -sin Ω • (ϕ + ω) sin Ω (ϕ • ) . (42)
We are going to show that ∂ ω K(ϕ, ω) is positive for ω > 0 sufficiently small. By the fundamental theorem of calculus, we have

∂ ω K(ϕ, ω) = sin Ω ((ϕ + ω) • ) sin Ω • (ϕ + ω) - ˆω 0 cos Ω ((ϕ + t) • ) -cos Ω (ϕ • ) dt + cos Ω ((ϕ + ω) • ) cos Ω • (ϕ + ω) + ˆω 0 sin Ω ((ϕ + t) • ) -sin Ω (ϕ • ) dt -cos Ω • (ϕ + ω) cos Ω (ϕ • ) -sin Ω • (ϕ + ω) sin Ω (ϕ • ) = 1 -cos Ω • (ϕ + ω) cos Ω (ϕ • ) -sin Ω • (ϕ + ω) sin Ω (ϕ • ) + ˆω 0 f (ϕ, ω, t) dt ≥ ˆω 0 f (ϕ, ω, t) dt,
where we have used the Pythagorean identity (3) and where f (t) := f (ϕ, ω, t) is given by

f (t) := cos Ω ((ϕ + ω) • ) sin Ω ((ϕ + t) • ) -sin Ω (ϕ • ) -sin Ω ((ϕ + ω) • ) cos Ω ((ϕ + t) • ) -cos Ω (ϕ • ) .
The integral ´ω 0 f (t) dt is positive, since again by (3),

∂ t f (t) = C ′ • (ϕ + t) sin Ω ((ϕ + ω) • ) sin Ω • (ϕ + t) + cos Ω ((ϕ + ω) • ) cos Ω • (ϕ + t)
=1 when ω=0 and t=0

, and thus by continuity ∂ t f (t) ≥ 0 for all ω > 0 sufficiently small and all t ∈ (0, ω). Since f (0) = 0, we deduce that f (t) ≥ 0, and ∂ ω K ≥ 0 for sufficiently small ω > 0. Then, since C • is non-increasing, the partial derivative in (42) is non-negative and therefore (41) holds. The second part of the statement follows from Propositions 3.23.

By using Lemma 5.1, we are able to give several necessary conditions for MCP to hold.

Theorem 5.2. Let H be the sub-Finsler Heisenberg group, equipped with a C 1 and strongly convex norm • , and with the Lebesgue measure L 3 . If MCP(0, N ) holds for some N ∈ (1, ∞), then the map

C ′ • : D 1 → R is continuous at every ϕ ∈ Z.
Proof. The claim is that if ϕ ∈ Z and if MCP(0, N ) holds, then C ′ • must be continuous at ϕ. Assume by contradiction this is not the case, then there exists a sequence {ϕ n } n∈N ⊆ D 1 such that ϕ n → ϕ as n → +∞ and lim n→+∞ C ′ • (ϕ n ) = C > 0. Now, setting ω n := ϕ nϕ and recalling that ϕ ∈ Z by assumption, we compute that

ω n C ′ • (ϕ + ω n ) C • (ϕ + ω n ) -C • (ϕ) = C ′ • (ϕ + ω n ) C•(ϕ+ωn)-C•(ϕ) ωn
→ +∞, as n → +∞.

By Lemma 5.1, the metric measure space (H, d, L 3 ) does not satisfy MCP(0, N ) for any N ∈ (1, ∞).

Theorem 5.3. Let H be the sub-Finsler Heisenberg group, equipped with a C 1 and strongly convex norm • , and with the Lebesgue measure L 3 . Assume that L 1 (Z) > 0. Then, the metric measure space (H, d, L 3 ) does not satisfy the MCP(0, N ) for any N ∈ (1, ∞).

Proof. Let 1 Z c be the indicator function of the complementary subset Z c ⊂ R/2π Ω • Z. By the Lebesgue differentiation theorem and the assumption L 1 (Z) > 0, there is a Lebesgue point

ϕ ⋆ ∈ Z of 1 Z c . Denote by D ϕ ⋆ the set of points ω such that ϕ ⋆ + ω ∈ D 1 . Note that, since the norm • is strongly convex, C • is Lipschitz (cf. Proposition 2.16
) and therefore for all δ > 0,

sup t∈(0,δ]∩D ϕ ⋆ C ′ • (ϕ ⋆ + t) < +∞.
Observe also that Lemma 3.15 ensures that this supremum is strictly positive. Then, for all δ > 0, there exists ω ∈ (0, δ] ∩ D ϕ ⋆ such that

C ′ • (ϕ ⋆ + ω) > 1 2 sup t∈(0,δ]∩D ϕ ⋆ C ′ • (ϕ ⋆ + t) ≥ 1 2 sup t∈(0,ω]∩D ϕ ⋆ C ′ • (ϕ ⋆ + t),
where the last inequality holds because (0, ω] ∩ D ϕ ⋆ ⊆ (0, δ] ∩ D ϕ ⋆ . In particular, there exists a sequence {ω n } n∈N ⊆ D ϕ ⋆ such that ω n → 0 as n → +∞, and

C ′ • (ϕ ⋆ + ω n ) > 1 2 sup ω∈(0,ωn]∩D ϕ ⋆ C ′ • (ϕ ⋆ + ω), ∀ n ∈ N. (43)
We make also the following simple observation:

1 ω n ˆωn 0 C ′ • (ϕ ⋆ + t) dt = 1 ω n ˆωn 0 C ′ • (ϕ ⋆ + t)1 Z c (ϕ ⋆ + t) dt ≤ 1 ω n ˆωn 0 1 Z c (ϕ ⋆ + t) dt sup ω∈(0,ωn]∩D ϕ ⋆ C ′ • (ϕ ⋆ + ω)
Consequently, we have the estimate

ω n C ′ • (ϕ ⋆ + ω n ) C • (ϕ ⋆ + ω n ) -C • (ϕ ⋆ ) = ω n C ′ • (ϕ + ω n ) ´ωn 0 C ′ • (ϕ ⋆ + ω) dω ≥ C ′ • (ϕ ⋆ + ω n ) 1 ωn ´ωn 0 1 Z c (ϕ ⋆ + t) dt sup ω∈(0,ωn]∩D ϕ ⋆ C ′ • (ϕ ⋆ + ω) > 1 2 ωn ´ωn 0 1 Z c (ϕ ⋆ + t) dt , (44)
where we have used (43) for the last inequality. Since ϕ ⋆ ∈ Z is a Lebesgue point of 1 Z c , we have

lim n→∞ 1 ω n ˆωn 0 1 Z c (ϕ ⋆ + t) dt = 1 Z c (ϕ ⋆ ) = 0.
Thus, taking the limit as n → +∞ in (44), we obtain

lim n→+∞ ω n C ′ • (ϕ ⋆ + ω n ) C • (ϕ ⋆ + ω n ) -C • (ϕ ⋆ ) = +∞.
This concludes the proof, as a consequence of Lemma 5.1.

By Theorems 5.2 and 5.3, the following conditions can always be assumed when investigating a sufficient condition for the measure contraction property:

L (Z) = 0 and C ′ • is continuous at every ϕ ∈ Z ( * )
However, as shown in the following theorem, the condition ( * ) does not give a sufficient condition for the measure contraction property. In particular, the MCP may fail due to a single zero point of C ′ • .

Theorem 5.4. Let H be the sub-Finsler Heisenberg group, equipped with a C 1 and strongly convex norm • , and equipped with the Lebesgue measure L 3 . Assume that there is ϕ ⋆ ∈ Z such that for any α ≥ 2,

C ′ • (ϕ) = o(|ϕ -ϕ ⋆ | α-2 ) as ϕ → ϕ ⋆ in D 1 . (45)
Then, the metric measure space (H, d, L 3 ) does not satisfy the MCP(0, N ) for any N ∈ (1, ∞).

Proof. We define the function β : R >0 → R ∪ {+∞} as follows:

β(ω) := log(C ′ • (ϕ ⋆ +ω)) log ω if ω ∈ D 1 ∩ Z c ∩ R >0 , +∞ otherwise.
In other words, β is such that

C ′ • (ϕ ⋆ + ω) = ω β(ω)
, whenever is finite. In addition, since the norm is strongly convex and L 1 (Z) = 0, β is finite almost everywhere. The assumption (45) implies that for any α ≥ 2, there is δ > 0 such that if 0 < ω < δ,

C ′ • (ϕ ⋆ + ω) ≤ ω α-2 .
Therefore, we have β(ω) ≥ α -2 for sufficiently small ω. Since α ∈ [2, +∞) is arbitrary, we deduce lim inf

ω→0 β(ω) = +∞. ( 46
)
We claim that there exists a sequence of positive numbers {ω n } n∈N with ω n → 0 such that

β(ω n ) < inf ω∈(0,ωn] β(ω) + log 2 log ω . (47)
Indeed, by (46), for any positive number ψ 1 ∈ D 1 , there is δ 1 = δ 1 (ψ 1 ) > 0 with the property that

β(ψ) < inf ω∈(0,ψ 1 ) β(ω) + 1 ⇒ |ψ| > δ 1 . (48)
Without loss of generality, we can choose δ 1 < 2. On the other hand, for any ε 1 > 0, there is

ω 1 ∈ (0, ψ 1 ) such that β(ω 1 ) < inf ω∈(0,ψ 1 ) β(ω) + ε 1 .
Choose ε 1 > 0 sufficiently small so that ε 1 < log 2 log δ 1 . Then, ω 1 satisfies

β(ω 1 ) < inf ω∈(0,ψ 1 ) β(ω) + ε 1 < inf ω∈(0,ψ 1 ) β(ω) + log 2 log δ 1 ≤ inf ω∈(0,ψ 1 ) β(ω) + log 2 log ω ≤ inf ω∈(0,ω 1 ] β(ω) + log 2 log ω ,
where the second-to-last inequality follows from (48). We can now repeat the same procedure starting from any ψ 2 < ω 1 to get a desired number ω 2 , and recursively get the desired sequence {ω n } n∈N which satisfies (47). Let β n := β(ω n ). By (47), for every ω ∈ (0, ω n ),

C ′ • (ϕ ⋆ + ω) ≤ 2ω βn and C ′ • (ϕ ⋆ + ω n ) = ω βn n .
Then, as n → ∞, we have:

ω n C ′ • (ϕ ⋆ + ω n ) C • (ϕ ⋆ + ω n ) -C • (ϕ ⋆ ) = ω βn+1 n ´ωn 0 C ′ • (ϕ ⋆ + ω) dω ≥ ω βn+1 n 2 ´ωn 0 ω βn dω = ω βn+1 n 2 βn+1 ω βn+1 n = β n + 1 2 → +∞.
By Lemma 5.1, the conclusion holds.

Remark 5.5. The condition (45) in Theorem 5.4 implies that the norm • is not C 1,α for any α ∈ (0, 1]. Indeed, by [Cio90, Prop. 1.6], the norm • is not C 1,α if and only if there is a sequence

{(ϕ n , ω n )} n∈N such that C • (ϕ n + ω n ) -C • (ϕ n ) ω 1/α n → 0 as n → ∞.
However, the converse is not necessarily true: if the norm • not being C 1,α for any α ∈ (0, 1] does not automatically imply the condition (45).

Sufficient condition by using C ′ •

In this section, we shall give a sufficient condition for the measure contraction property. We start from rewriting the reduced Jacobian function J R in terms the function C • . Recall that the differential equality sin ′′ Ω

• (x) = -C ′ • (x) sin Ω • (x) holds for x ∈ D 1 .
Since D 1 is a full measure subset, we are allowed to use the integration by parts to write

sin Ω • (x) = sin Ω • (y) + ˆx y cos Ω (t • ) dt = sin Ω • (y) + cos Ω (y • )(x -y) - ˆx y sin Ω • (t)C ′ • (t)(x -t) dt, (49)
which is the Taylor's formula with integral remainder of order 2. Note that the integrand of the last term of (49) contains sin Ω • (t), and it can be replaced recursively by the same expression (49), i.e.

sin

Ω • (x) = sin Ω • (y) + cos Ω (y • )(x -y) - ˆx y sin Ω • (y) + cos Ω (y • )(t -y) - ˆt y sin Ω • (s)C ′ • (s)(t -s) dt C ′ • (t)(x -t) dt = sin Ω • (y) + cos Ω (y • )(x -y) -sin Ω • (y) ˆx y C ′ • (t)(x -t) dt -cos Ω (y • ) ˆx y C ′ • (t)(x -t)(t -y) dt + ˆx y ˆt y sin Ω • (s)C ′ • (t)C ′ • (s)(x -t)(t -s) ds dt.
We repeat this process a last time, that is to say, we replace the term sin Ω • (s) in the last integral term above by its expression (49):

sin Ω • (x) = sin Ω • (y) + cos Ω (y • )(x -y) -sin Ω • (y) ˆx y C ′ • (t)(x -t) dt -cos Ω (y • ) ˆx y C ′ • (t)(x -t)(t -y) dt + sin Ω • (y) ˆx y ˆt y C ′ • (s)C ′ • (t)(x -t)(t -s) ds dt + cos Ω (y • ) ˆx y ˆt y C ′ • (s)C ′ • (t)(x -t)(t -s)(s -y) ds dt - ˆx y ˆt y ˆs y sin Ω • (u)C ′ • (t)C ′ • (s)C ′ • (u)(x -t)(t -s)(s -u) du ds dt. (50)
The same reasoning can be done for cos

Ω • (x), starting from cos ′′ Ω • (x) = -C ′ • (x) cos Ω • (x)
with similar computations and we get

cos Ω • (x) = cos Ω • (y) -sin Ω (y • )(x -y) -cos Ω • (y) ˆx y C ′ • (t)(x -t) dt + sin Ω (y • ) ˆx y C ′ • (t)(x -t)(t -y) dt + cos Ω • (y) ˆx y ˆt y C ′ • (s)C ′ • (t)(x -t)(t -s) ds dt -sin Ω (y • ) ˆx y ˆt y C ′ • (s)C ′ • (t)(x -t)(t -s)(s -y) ds dt - ˆx y ˆt y ˆs y cos Ω • (u)C ′ • (t)C ′ • (s)C ′ • (u)(x -t)(t -s)(s -u) du ds dt. ( 51
)
Since cos Ω (x • ) = d dx sin Ω • (x) and sin Ω (x • ) = -d dx cos Ω • (x), we also obtain

cos Ω (x • ) = cos Ω (y • ) -sin Ω • (y) ˆx y C ′ • (t) dt -cos Ω (y • ) ˆx y C ′ • (t)(t -y) dt + sin Ω • (y) ˆx y ˆt y C ′ • (s)C ′ • (t)(t -s) ds dt + cos Ω (y • ) ˆx y ˆt y C ′ • (s)C ′ • (t)(t -s)(s -y) ds dt - ˆx y ˆt y ˆs y sin Ω • (u)C ′ • (t)C ′ • (s)C ′ • (u)(t -s)(s -u) du ds dt, (52)
and

sin Ω (x • ) = sin Ω (y • ) + cos Ω • (y) ˆx y C ′ • (t) dt -sin Ω (y • ) ˆx y C ′ • (t)(t -y) dt -cos Ω • (y) ˆx y ˆt y C ′ • (s)C ′ • (t)(t -s) ds dt + sin Ω (y • ) ˆx y ˆt y C ′ • (s)C ′ • (t)(t -s)(s -y) ds dt + ˆx y ˆt y ˆs y cos Ω • (u)C ′ • (t)C ′ • (s)C ′ • (u)(t -s)(s -u) du ds dt. ( 53
)
The combination of these formulas leads to the following result.

Proposition 5.6. Let H be the sub-Finsler Heisenberg group, equipped with a C 1 and strongly convex norm • . Then, its reduced Jacobian J R can be expressed in the following way. For all (ϕ, ω) ∈ U , it holds

J R (ϕ, ω) = 1 2 ˆϕ+ω ϕ ˆϕ+ω ϕ (t -s) 2 C ′ • (t)C ′ • (s) ds dt + R(ϕ, ω), (54)
and, if in addition ϕ + ω ∈ D 1 , it holds:

∂ ω J R (ϕ, ω) = C ′ • (ϕ + ω) ˆϕ+ω ϕ (ϕ + ω -t) 2 C ′ • (t) dt + ∂ ω R(ϕ, ω), (55)
where the remainder term R(ϕ, ω) is given by

R(ϕ, ω) = ˆϕ+ω ϕ ˆt ϕ ˆs ϕ (t -s)(s -u) sin Ω • (u) cos Ω • (ϕ) -sin Ω • (ϕ) cos Ω • (u) -(t -ϕ)(cos Ω (ϕ • ) cos Ω • (u) + sin Ω (ϕ • ) sin Ω • (u)) C ′ • (t)C ′ • (s)C ′ • (u) du ds dt, (56)
and its derivative by

∂ ω R(ϕ, ω) = C ′ • (ϕ + ω) ˆϕ+ω ϕ ˆs ϕ (ϕ + ω -s)(s -u) sin Ω • (u) cos Ω • (ϕ) -sin Ω • (ϕ) cos Ω • (u) -ω(cos Ω (ϕ • ) cos Ω • (u) + sin Ω (ϕ • ) sin Ω • (u)) C ′ • (s)C ′ • (u) du ds. ( 57
)
Remark 5.7. This formula is a synthetic generalisation of the expansion of the Jacobian that appeared in [START_REF] Borza | Measure contraction property, curvature exponent and geodesic dimension of sub-Finsler Heisenberg groups[END_REF]Lem. 29] and [BT23, Eq. ( 30)] for the ℓ p -Heisenberg group.

Proof of Proposition 5.6. The equality (54) is shown by substituting (50), (51), (52), and (53), with y = ϕ and x = ϕ+ω, into the expression of the reduced Jacobian (14). The term sin Ω • (ϕ+ω), for example, is replaced using (50) by

sin Ω • (ϕ + ω) = sin Ω • (ϕ) + ω cos Ω (ϕ • ) (58) -sin Ω • (ϕ) ˆϕ+ω ϕ C ′ • (t)(ϕ + ω -t) dt -cos Ω (ϕ • ) ˆϕ+ω ϕ C ′ • (t)(ϕ + ω -t)(t -ϕ) dt (59) + sin Ω • (ϕ) ˆϕ+ω ϕ ˆt ϕ C ′ • (s)C ′ • (t)(ϕ + ω -t)(t -s) ds dt (60) + cos Ω (ϕ • ) ˆϕ+ω ϕ ˆt ϕ C ′ • (s)C ′ • (t)(ϕ + ω -t)(t -s)(s -ϕ) ds dt (61) - ˆϕ+ω ϕ ˆt ϕ ˆs ϕ sin Ω • (r)C ′ • (t)C ′ • (s)C ′ • (r)(ϕ + ω -t)(t -s)(s -r) dr ds dt. ( 62
)
This is also done for the terms cos Ω • (ϕ + ω), sin Ω ((ϕ + ω) • ), and cos Ω ((ϕ + ω) • ) in (14). After a computation, one can find that all the terms of "order zero" (those with no integral and no C ′ • term) such as (58) vanish. All the terms of "order one" (those with one integral and one C ′

• term) such as (59) cancel out too. The terms of "second order" (those with a double integral and two C ′ • terms) such as (60)-(61), however, do not cancel and they simplify to the first term in (54). The remainder (56) is obtained by gathering all the terms of "higher order" (those with three integrals and three C ′

• terms) such as (62). Finally, differentiating with respect to ω yields (55) and (57).

Remark 5.8. Since the generalised trigonometric functions are bounded and C • is Lipschitz, there is a constant C > 0 such that for all (ϕ, ω) ∈ U , we have

|R(ϕ, ω)| ≤ C ˆϕ+ω ϕ ˆt ϕ ˆs ϕ (t -s)(s -u)C ′ • (t)C ′ • (s)C ′ • (u) du ds dt ≤ C ˆϕ+ω ϕ ˆϕ+ω ϕ ˆϕ+ω ϕ (t -u) 2 C ′ • (t)C ′ • (s)C ′ • (u) du ds dt ≤ 2C C ′ • ∞ ω 1 2 ˆϕ+ω ϕ ˆϕ+ω ϕ (t -s) 2 C ′ • (t)C ′ • (s) ds dt (63)
and, similarly, when defined,

|∂ ω R(ϕ, ω)| ≤ 2C C ′ • ∞ ω C ′ • (ϕ + ω) ˆϕ+ω ϕ (ϕ + ω -t) 2 C ′ • (t) dt (64)
with an appropriate constant C > 0.

To simplify the notation in the proof of the next theorem, we introduce the following functions. For α ≥ 2, we define

P α (ϕ, ω) := 1 2 ˆϕ+ω ϕ ˆϕ+ω ϕ (t -s) 2 |t| α-2 |s| α-2 ds dt = 1 α 2 (α 2 -1) |ω + ϕ| 2α + |ϕ| 2α ) + 2 α 2 |ω + ϕ| α |ϕ| α - 1 α 2 -1 |ω + ϕ| α |ϕ| α-2 + |ω + ϕ| α-2 |ϕ| α (ω + ϕ)ϕ, (65)
and P α (s) := P α (1, s). Note that

∂ ω P α (ϕ, ω) = 2 α(α 2 -1) |ω + ϕ| 2(α-1) (w + ϕ) + 2 α |ω + ϕ| α-2 (ω + ϕ)|ϕ| α - 1 α 2 -1 (α + 1)|ω + ϕ| 2 + (α -1)|ϕ| 2 |ω + ϕ| α-2 |ϕ| α-2 ϕ,
and that ∂ s P α (s) = ∂ ω P α (1, s). More explicitly, the functions P α (s) and ∂ s P α (s) are given by

P α (s) := 1 α 2 (α 2 -1) |1 + s| 2α -α 2 |1 + s| α (1+s)+2(α 2 -1)|1 + s| α -α 2 |1 + s| α-2 (1+s)+1 , (66)
and

∂ s P α (s) = 2 α(α 2 -1) |1 + s| α-2 |1 + s| α (1 + s) -1 -(α + 1)s - α(α + 1) 2 s 2 .
The function P α (s) vanishes if and only if s = 0. Indeed, if 1 + s < 0, then it is easy to see that all the terms of P α (s) are strictly positive. If 1 + s > 0, the derivative of P α (s) is

∂ s P α (s) = 2α(1 + s) α-2 (1 + s) α+1 -1 -(α + 1)s - α(α + 1) 2 s 2 .
Consider the function f (s) := (1 + s) α+1 -1 -(α + 1)s -α(α+1) 2 s 2 . Then f ′′ (s) = α(α + 1) |1 + s| α-1 -1 , thus f ′ (s) is convex on (-1, +∞) with a unique minimum f ′ (0) = 0 and therefore f is increasing on (-1, +∞). Since f (-1) = α(1-α) 2 < 0 and f (0) = 0, ∂ s P α (s) is increasing with a unique zero point ∂ s P α (0) = 0. Therefore, P α (s) attains a unique minimum at 0. Since P α (0) = 0 we conclude that s = 0 is the unique zero point of P α . The map s → 1 + s∂sPα(s) Pα(s) is thus continuous for all s ∈ R and bounded.

Theorem 5.9. Let H be the sub-Finsler Heisenberg group, equipped with a C 1 and strongly convex norm • . Assume that for all ϕ ⋆ ∈ R/2π Ω• Z, there exists α(ϕ ⋆ ) ∈ [2, +∞) and A(ϕ ⋆ ), B(ϕ ⋆ ) > 0 such that

A(ϕ ⋆ )|ϕ -ϕ ⋆ | α(ϕ ⋆ )-2 ≤ C ′ • (ϕ) ≤ B(ϕ ⋆ )|ϕ -ϕ ⋆ | α(ϕ ⋆ )-2 for a.e. ϕ near ϕ ⋆ . ( 68
)
Assume furthermore that

sup ϕ ⋆ ∈R/2π Ω• Z B(ϕ ⋆ ) A(ϕ ⋆ ) , sup ϕ ⋆ ∈R/2π Ω• Z α(ϕ ⋆ ) < +∞. ( 69
)
Then, the metric measure space (H, d, L 3 ) satisfies the MCP(0, N ) condition for some N ∈ (1, ∞).

Proof. Fix ϕ ⋆ ∈ R/2π Ω • Z. If (ϕ, ω) ∈ U is close enough to (ϕ ⋆ , 0
), then we have, by (54) and the assumption, that

J R (ϕ, ω) = 1 2 ˆϕ+ω ϕ ˆϕ+ω ϕ (t -s) 2 C ′ • (t)C ′ • (s) ds dt + R(ϕ, ω) ≥ A(ϕ ⋆ ) 2 2 ˆϕ+ω ϕ ˆϕ+ω ϕ (t -s) 2 |t -ϕ ⋆ | α(ϕ ⋆ )-2 |s -ϕ ⋆ | α(ϕ ⋆ )-2 ds dt + R(ϕ, ω) = A(ϕ ⋆ ) 2 P α(ϕ ⋆ ) (ϕ -ϕ ⋆ , ω) + R(ϕ, ω) ≥ A(ϕ ⋆ ) 2 -CB(ϕ ⋆ ) 2 ω P α(ϕ ⋆ ) (ϕ -ϕ ⋆ , ω),
where P α(ϕ ⋆ ) (ϕ, ω) is the function defined in (65) and a lower bound on the remainder term R(ϕ, ω) can be obtained from (63) and the lower bound of (68), with C := 2C C ′ • ∞ . Similarly, we can deduce an upper bound for ∂ ω J R from (55) and the upper bound of (68):

∂ ω J R (ϕ, ω) = C ′ • (ϕ + ω) ˆϕ+ω ϕ (ϕ + ω -t) 2 C ′ • (t) dt + ∂ ω R(ϕ, ω) ≤ B(ϕ ⋆ ) 2 ∂ ω P α(ϕ ⋆ ) (ϕ -ϕ ⋆ , ω) + ∂ ω R(ϕ, ω) ≤ B(ϕ ⋆ ) 2 + CB(ϕ ⋆ ) 2 ω ∂ ω P α(ϕ ⋆ ) (ϕ -ϕ ⋆ , ω),
where we also have a bound for the remainder term ∂ ω R(ϕ, ω) using (64). The goal now is to show that lim sup (ϕ,ω)→(ϕ ⋆ ,0) N (ϕ, ω) has a uniform upper bound independent of ϕ ⋆ , since this would prove that the MCP(0, N ) condition is satisfied for some N ∈ (1, ∞), according to Proposition 3.23. A bound for lim sup (ϕ,ω)→(ϕ ⋆ ,0) N (ϕ, ω) is found following the blueprint of the proof of [START_REF] Borza | Measure contraction property, curvature exponent and geodesic dimension of sub-Finsler Heisenberg groups[END_REF]Thm. 39]. Consider a converging sequence {(ϕ n , ω n )} n∈N ⊂ U such that ϕ n + ω n ∈ D + 1 and (ϕ n , ω n ) → (ϕ ⋆ , 0) as n → +∞. Assume also, without loss of generality, that that the ratio s n := ω n /(ϕ n -ϕ ⋆ ) converges to s ∈ [-∞, +∞] as n → +∞. Since ω n → 0, we obtain a positive lower bound

J R (ϕ n , ω n ) ≥ A(ϕ ⋆ ) 2 -CB(ϕ ⋆ ) 2 ω n P α(ϕ ⋆ ) (ϕ n -ϕ ⋆ , ω n ) = A(ϕ ⋆ ) 2 -CB(ϕ ⋆ ) 2 ω n |ϕ n -ϕ ⋆ | 2α(ϕ ⋆ ) P α(ϕ ⋆ ) (s n ).
We estimate ω n J R (ϕ n , ω n ) is a similar fashion:

ω n ∂ ω J R (ϕ n , ω n ) ≤ B(ϕ ⋆ ) 2 + CB(ϕ ⋆ ) 2 ω n ∂ ω P α(ϕ ⋆ ) (ϕ n -ϕ ⋆ , ω n ) = B(ϕ ⋆ ) 2 + CB(ϕ ⋆ ) 2 ω n |ϕ n -ϕ ⋆ | 2α(ϕ ⋆ ) s n ∂ s P α(ϕ ⋆ ) (s n ).
With these bounds in hand, we can write

lim sup n→∞ N (ϕ n , ω n ) ≤ lim n→∞ 1 + B(ϕ ⋆ ) 2 + CB(ϕ ⋆ ) 2 ω n A(ϕ ⋆ ) 2 -CB(ϕ ⋆ ) 2 ω n • |ϕ n -ϕ ⋆ | 2α(ϕ ⋆ ) |ϕ n -ϕ ⋆ | 2α(ϕ ⋆ ) • s n ∂ s P α(ϕ ⋆ ) (s n ) P α(ϕ ⋆ ) (s n ) = 1 + B(ϕ ⋆ ) 2 A(ϕ ⋆ ) 2 s∂ s P α(ϕ ⋆ ) (s) P α(ϕ ⋆ ) (s) ≤ 1 + sup ϕ ⋆ ∈R/2π Ω • Z B(ϕ ⋆ ) 2 A(ϕ ⋆ ) 2 sup s∈[-∞,+∞] sup ϕ ⋆ ∈R/2π Ω • Z s∂ s P α(ϕ ⋆ ) (s) P α(ϕ ⋆ ) (s) .
Taking into account the first assumption in (69), it is enough to show that the second supremum is finite. Since the function

(α, s) → s∂ s P α (s) P α (s)
is continuous and bounded as s → ±∞ (cf. ( 67)) and thus uniformly bounded on [2, M ] × [-∞, +∞] for every M ∈ (2, +∞), the second assumption in (69) allows to conclude.

The next theorem shows that, we can get a lower bound on the curvature exponent, cf. Definition 2.6, by assuming C ′

• is differentiable in a fractional way.

Theorem 5.10. Let H be the sub-Finsler Heisenberg group, equipped with a C 1 and strongly convex norm • . Assume that for all ϕ ⋆ ∈ R/2π Ω• Z, there exists α(ϕ ⋆ ) ∈ [2, +∞) and A(ϕ ⋆ ) > 0 such that

C ′ • (ϕ) = A(ϕ ⋆ )|ϕ -ϕ ⋆ | α(ϕ ⋆ )-2 + o(|ϕ -ϕ ⋆ | α(ϕ ⋆ )-2 ), as ϕ → ϕ ⋆ in D 1 . (70)
Assume furthermore that q := sup{α(ϕ

⋆ ) | ϕ ⋆ ∈ R/2π Ω• Z} < +∞.
Then, the metric measure space (H, d, L 3 ) satisfies the MCP(0, N ) condition for some N ∈ (1, +∞). In addition, it holds that

N curv ≥ 2q + 1.
Proof. Observe that the assumptions of Theorem 5.9 are verified, therefore (H, d, L 3 ) satisfies the MCP(0, N ) condition for some N ∈ (1, +∞). In the reminder of the proof we find an estimate for its curvature exponent N curv . The argument follows the same lines as the proof of Theorem 5.9, but with finer asymptotics. Let ϕ ⋆ ∈ R/2π 2Ω • Z and, for simplicity, we write A = A(ϕ ⋆ ). We start by showing that

J R (ϕ, ω) = A 2 2 ˆϕ+ω ϕ ˆϕ+ω ϕ (t -s) 2 |t -ϕ ⋆ | α(ϕ ⋆ )-2 |s -ϕ ⋆ | α(ϕ ⋆ )-2 ds dt =A 2 P α(ϕ ⋆ ) (ϕ-ϕ ⋆ ,ω) +o(P α(ϕ ⋆ ) (ϕ -ϕ ⋆ , ω)),
as (ϕ, ω) → (ϕ ⋆ , 0). In order to prove that, consider the leading term of (54), and observe that 1 2

ˆϕ+ω ϕ ˆϕ+ω ϕ (t -s) 2 C ′ • (t)C ′ • (s) ds dt = A 2 2 ˆϕ+ω ϕ ˆϕ+ω ϕ (t -s) 2 |t -ϕ ⋆ | α(ϕ ⋆ )-2 |s -ϕ ⋆ | α(ϕ ⋆ )-2 ds dt + 1 2 ˆϕ+ω ϕ ˆϕ+ω ϕ (t -s) 2 A|t -ϕ ⋆ | α(ϕ ⋆ )-2 (C ′ • (s) -A|s -ϕ ⋆ | α(ϕ ⋆ )-2 ) ds dt (71) + 1 2 ˆϕ+ω ϕ ˆϕ+ω ϕ (t -s) 2 (C ′ • (t) -A|t -ϕ ⋆ | α(ϕ ⋆ )-2 )A|s -ϕ ⋆ | α(ϕ ⋆ )-2 ds dt (72) + 1 2 ˆϕ+ω ϕ ˆϕ+ω ϕ (t -s) 2 (C ′ • (t) -A|t -ϕ ⋆ | α-2 )(C ′ • (s) -A|s -ϕ ⋆ | α(ϕ ⋆ )-2 ) ds dt. ( 73
)
The term (71) is o(P α(ϕ ⋆ ) (ϕϕ ⋆ , ω)) as (ϕ, ω) → (ϕ ⋆ , 0). Indeed, by (70), for all ε > 0, we have that

1 2 ˆϕ+ω ϕ ˆϕ+ω ϕ (t -s) 2 (C ′ • (t) -A|t -ϕ ⋆ | α(ϕ ⋆ )-2 )(C ′ • (s) -A|s -ϕ ⋆ | α(ϕ ⋆ )-2 ) ds dt ≤ ε 2 A 2 2 ˆϕ+ω ϕ ˆϕ+ω ϕ (t -s) 2 |t -ϕ ⋆ | α(ϕ ⋆ )-2 |s -ϕ ⋆ | α(ϕ ⋆ )-2 ds dt, for all (ϕ, ω) ∈ U sufficiently close to (ϕ ⋆ , 0) so that C ′ • (t) -A|t -ϕ ⋆ | α(ϕ ⋆ )-2 ≤ ε|t -ϕ ⋆ | α(ϕ ⋆ )-2 , for all t ∈ [ϕ, ϕ + ω].
Analogous computations show that the terms (72), (73) are o(P α(ϕ ⋆ ) (ϕϕ ⋆ , ω)) as (ϕ, ω) → (ϕ ⋆ , 0). The same thing can be done for ∂ ω J R (ϕ, ω) with the help of (55) and (64). More precisely, under the condition (70), we have that

∂ ω J R (ϕ, ω) = A 2 |ϕ + ω -ϕ ⋆ | α(ϕ ⋆ )-2 ˆϕ+ω ϕ (ϕ + ω -t) 2 |t -ϕ ⋆ | α(ϕ ⋆ )-2 dt =A 2 ∂ωP α(ϕ ⋆ ) (ϕ-ϕ ⋆ ,ω) +o(∂ ω P α(ϕ ⋆ ) (ϕ-ϕ ⋆ , ω)), as (ϕ, ω) → (ϕ ⋆ , 0). Now, given any s ∈ [-∞, +∞], consider a sequence {(ϕ n , ω n )} n∈N ⊂ U such that ϕ n , ϕ n +ω n ∈ D 1 , (ϕ n , ω n ) → (ϕ ⋆ , 0) as n → +∞ and the ratio s n := ω n /(ϕ n -ϕ ⋆ ) converges to s ∈ [-∞, +∞] as n → +∞. Then, we obtain that, as n → ∞, N (ϕ n , ω n ) = 1 + A 2 ω n ∂ ω P α(ϕ ⋆ ) (ϕ n -ϕ ⋆ , ω n ) + o(ω n ∂ ω P α(ϕ ⋆ ) (ϕ n -ϕ ⋆ , ω n )) A 2 P α(ϕ ⋆ ) (ϕ n -ϕ ⋆ , ω n ) + o(P α(ϕ ⋆ ) (ϕ n -ϕ ⋆ , ω n )) = 1 + |ϕ n -ϕ ⋆ | 2α(ϕ ⋆ ) A 2 s n ∂ ω P α(ϕ ⋆ ) (s n ) + s n o(∂ s P α(ϕ ⋆ ) (s n )) |ϕ n -ϕ ⋆ | 2α(ϕ ⋆ ) A 2 P α(ϕ ⋆ ) (s n ) + o(P α(ϕ ⋆ ) (s n )) → 1 + s∂ s P α(ϕ ⋆ ) (s) P α(ϕ ⋆ ) (s) , (74)
where P α(ϕ ⋆ ) (s) is the function defined in (66). Keeping in mind (74) and since s ∈ [-∞, +∞] is arbitrary, we finally deduce that sup

ϕ ⋆ ∈R/2π Ω • Z lim sup (ϕ,ω)→(ϕ ⋆ ,0) N (ϕ, ω) = sup ϕ ⋆ ∈R/2π Ω • Z sup s∈[-∞,+∞] 1 + s∂ s P α(ϕ ⋆ ) (s) P α(ϕ ⋆ ) (s) ≥ 2q + 1,
where the last inequality is a consequence of the estimate that lim s→±∞ s∂sPq(s) Pq(s) = 2q cf. (67). The conclusion follows from Corollary 3.22.

Remark 5.11. (i) If q > 2 and there is ϕ ⋆ such that α(ϕ ⋆ ) = q, then one can prove that the curvature exponent satisfies N curv > 2q + 1. Indeed, it is possible to observe that, in this case, the map s → 1 + s∂ s P q (s) P q (s) converges from above to its limits at -∞.

(ii) For p ∈ (1, 2), consider the ℓ p -Heisenberg group, that is to say the sub-Finsler Heisenberg group equipped with an ℓ p -norm. The ℓ p -norm is C 1 and strongly convex, and its angle correspondence C • satisfies

C ′ • (ϕ) = (q -1)|ϕ -ϕ ⋆ | q-2 + o(|ϕ -ϕ ⋆ | q-2 ) if ϕ ⋆ ∈ πq 2 Z (q -1)| sin ℓ q (ϕ ⋆ ) cos ℓ q (ϕ ⋆ )| q-2 + o(1) if ϕ ⋆ / ∈ πq 2 Z , as ϕ → ϕ ⋆ .
Therefore, Theorem 5.10, together with item (i), recovers the estimate in [START_REF] Borza | Measure contraction property, curvature exponent and geodesic dimension of sub-Finsler Heisenberg groups[END_REF].

As a consequence of the previous results, we also obtain the following important corollary.

Corollary 5.12. Let H be the sub-Finsler Heisenberg group, equipped with a C 1,1 and strongly convex norm • . Then, the metric measure space (H, d, L 3 ) satisfies MCP(0, N ) for some

N ∈ (1, ∞). Furthermore, if the norm • is C 2 , then N curv ≥ 5.
Proof. As the norm • is C 1,1 and strongly convex, then C • and C • are Lipschitz continuous, cf. Proposition 2.16. In particular, for every ϕ

∈ D 1 , 1/A ≤ C ′ • (ϕ) ≤ A for some constant A > 0 since C • • C • = Id.
Therefore, the hypothesis of Theorem 5.9 are satisfied with α(ϕ ⋆ ) = 2 for every ϕ ⋆ ∈ R/2π Ω• Z, and the first claim follows.

If the norm is C 2 , then the (inverse) angle correspondence map C • is of class C 1 . By the inverse function theorem, C • is also of class C 1 , and C ′ • is continuous. Then, Theorem 5.10 holds with q = 2. This implies the last part of the claim.

Remark 5.13. (i) For the estimate N ≥ 5, we only need the continuity of C ′

• at one non-zero point. However, if the norm is only C 1,1 , the existence of such a point is not guaranteed.

(ii) If the norm is C 2 , then what we have proven is that lim sup

(ϕ,ω)→(ϕ ⋆ ,0) N (ϕ, ω) = 5
for every ϕ ⋆ ∈ R/2π Ω • (and thus N curv ≥ 5). To prove MCP(0, 5) (so that N curv = 5), we would need to show the inequality N (ϕ, ω) ≤ 5 for ω = 0.

The following example shows that there is a C 2 and strongly convex norm such that the associated sub-Finsler Heisenberg group has N curv > 5.

Example 5.14. For K, L > 0 and ε ∈ (0, K/L), let f be the function defined on (-ε, ε) by

f (ϕ) = K + Lϕ.
Since f is bounded, positive and smooth, there is a C 2 and strongly convex norm on R 2 such that its angle correspondence satisfies C • (ϕ) = ´ϕ 0 f (t) dt in a sufficiently small neighbourhood of ϕ = 0, and such that cos Ω (0 • ) = cos Ω • (0) = 1, sin Ω (0 • ) = sin Ω • (0) = 0 (for the construction of such a norm, see Remark 5.15). For ω ∈ (-ε, ε), we have that

1 2 ˆω 0 ˆω 0 (t -s) 2 C ′ • (t)C ′ • (s) ds dt = 1 2 ˆω 0 ˆω 0 (t -s) 2 (K + Ls)(K + Lt) ds dt = ω 4 12 K 2 + KLω + L 2 6 ω 2 .
By (56) and the assumption on the initial values, the remainder term R(0, ω) satisfies

R(0, ω) = ˆω 0 ˆt 0 ˆs 0 (t -s)(s -u)(u -t + O(u 2 ))C ′ • (t)C ′ • (s)C ′ • (u) du ds dt = o(ω 6 ) as ω → 0.
Therefore, by (54), we have that

J R (0, ω) = ω 4 12 K 2 + KLω + L 2 6 ω 2 + o(ω 6 ) as ω → 0.
Since ω ∈ D 1 by construction, we also have that

ωC ′ • (ω) ˆω 0 (ω -t) 2 C ′ • (t) dt = ω(K + Lω) ˆω 0 (ω -t) 2 (K + Lt) dt = 1 3 ω 4 K 2 + 5KL 4 ω + L 2 ω 2 4 .
In addition, by (57), it holds that ω∂ ω R(0, ω) = o(ω 6 ), as ω → 0. Thus, recalling (55), we have:

ω∂ ω J R (0, ω) = 1 3 ω 4 K 2 + 5KL 4 ω + L 2 ω 2 4 + o(ω 6 ).
Consequently, for sufficiently small ω > 0, we obtain that

4J R (0, ω) -ω∂ ω J R (0, ω) = ω 4 3 - KLω 4 - L 2 ω 2 12 + o(ω 6 ) < 0.
This shows that 1 + ω∂ ω J R (0, ω)/J R (0, ω) > 5 and N curv > 5 by Proposition 3.22.

Remark 5.15. For a given integrable, bounded and almost everywhere positive function f , we construct a norm • whose angle correspondence C • satisfies the following conditions:

C • is Lipschitz continuous and strictly increasing, C ′ • (ϕ) = f (ϕ) near ϕ = 0. (75)
Recall that the generalized trigonometric functions satisfy the differential equations

d dt cos Ω • (t) = -sin Ω (t • ), d dt sin Ω (t • ) = C ′ • (t) cos Ω • (t), d dt cos Ω (t • ) = -C ′ • (t) sin Ω • (t), d dt sin Ω • (t) = cos Ω (t • ),
provided that the norm • is C 1 and strongly convex. Conversely, the following differential equations locally recovers a generalized trigonometric function:

ẋ(t) = -Y (t), Ẏ (t) = f (t)x(t), Ẋ(t) = -f (t)y(t), ẏ(t) = X(t). (76)
Indeed, by Carathéodory's Theorem, there is an absolutely continuous solution to the differential equation (76) for a given initial value (X 0 , Y 0 , x 0 , y 0 ) at time t = 0. Let (X, Y, x, y) be a solution to the initial value (1, 0, 1, 0). Note that, since (X, Y ) is absolutely continuous and its differential is bounded, the curve (x, y) is C 1,1 . Then, we have obtained C 1,1 curves γ ± : (-ε, ε) → R 2 , γ + (t) := (x(t), y(t)) and γ -(t) := (-x(t), -y(t)). Without loss of generality, we can assume that γ ± is twice differentiable at endpoints. Join γ + (ε) and γ -(-ε) by an Euclidean arc smoothly. The central inversion of this arc joins γ + (-ε) and γ -(ε). This procedure yields a centrally symmetric, strictly convex and C 1,1 domain B, and define • * as the norm whose unit ball is B. Let • be the dual norm of • * . From the construction, the angle correspondence C • of the norm • satisfies the condition (75).

Towards a necessary and sufficient condition

In this section, we provide examples of norms that satisfy and do not satisfy the measure contraction property. We begin by presenting an example of a norm that satisfies the MCP condition, even though it is not included by the assumptions of Theorem 5.9.

Example 5.16. Let f be defined by f (ϕ) = |ϕ log |ϕ|| in a neighbourhood of ϕ = 0. Since f is bounded, strictly positive and locally integrable, following the construction laid out in Remark 5.15, there is a C 1 and strongly convex norm on R 2 such that its angle correspondence satisfies C • (ϕ) = ´ϕ 0 f (t) dt in a sufficiently small neighbourhood of ϕ = 0. Note that C ′

• is everywhere positive except at ϕ = 0 or ϕ = π Ω • . We shall see that the sub-Finsler Heisenberg group associated with this norm satisfies the measure contraction property.

For sufficiently small and positive ϕ and ω, we have that Since the only zero points of C ′ • are ϕ = 0 and ϕ = π Ω • and lim sup (ϕ,ω)→(ϕ ⋆ ,0) N (ϕ, ω) = 5 for the other values of ϕ ⋆ , proving that lim sup (ϕ,ω)→(0,0) N (ϕ, ω) < +∞ implies that the corresponding sub-Finsler Heisenberg group satisfies MCP, according to Proposition 3.23 (it follows by symmetry that lim sup (ϕ,ω)→(π Ω • ,0) N (ϕ, ω) < +∞).

1 2 ˆϕ+ω ϕ ˆϕ+ω ϕ (s -t) 2 C ′ • (s)C ′ • (
In Example 5.16, we have seen that if the angle correspondence satisfies C ′ • ∼ |ϕ log |ϕ|| near its zero points, then it is possible to verify the measure contraction property. This means that the sufficient condition of Theorem 5.9 is not a necessary condition. Indeed, the upper bound of C ′

• in (68) is satisfied with α ≤ 3 while the lower bound is only satisfied with α > 3. In some sense, this indicates that the principal term of C ′ • need not have fractional order at any zero point for the MCP to hold.

Next, we present an example of sub-Finsler structure on the Heisenberg group for which the the differential of the angle correspondence C ′

• oscillates between |ϕ| and |ϕ log |ϕ||. This example is going to tell us that if the differential of the angle correspondence oscillates, then the measure contraction property may fail.

Example 5.17. Let {a n } n∈N , {b n } n∈N and {c n } n∈N be three sequences of positive real numbers converging to zero such that a n < b n < c n , c n+1 = a n , and a n /b n → 0 as n → +∞. In particular, b n /(b na n ) → 1 and a n /(b na n ) → 0 as n → +∞. Let f be the function defined almost everywhere by

f (ϕ) = |ϕ| if |ϕ| ∈ (a n , b n ) |ϕ log |ϕ|| if |ϕ| ∈ (b n , c n ).
Since f is bounded, positive and locally integrable, there is a C 1 and strongly convex norm on R 2 such that its angle correspondence satisfies C • (ϕ) = ´ϕ 0 f (t) dt in a sufficiently small neighbourhood of ϕ = 0 as in Remark 5.15. Set ϕ n := a n and ω n := b n + ε na n , where ε n > 0 is chosen so that b n + ε n ∈ (b n , c n ) and ε n ≤ (b na n ) 3 . We have that

n )ε n (ε n + b n -a n ) 2 b n (b n + ε n )| log b n | (b n -a n ) 6 ,
and the right-hand side tends to zero as n → +∞. A similar computation can be done for the other terms in (78) and (79). Since ϕ n + ω n ∈ D 1 by construction, we also have that Note that, by construction, we have that α(ϕ ⋆ ) ≥ β(ϕ ⋆ ). Intuitively, they are the optimal constants for which following inequality holds asymptotically

ω n C ′ • (ϕ n + ω n ) ˆϕn+ωn ϕn (ϕ n + ω n -t) 2 C ′ • (t) dt = -(b n -a n + ε n )(b n + ε n ) log(b n + ε n ) ˆbn an (b n + ε n -t) 2 t dt - ˆbn+εn bn (b n + ε n -t) 2 t log t dt = -(b n -a n + ε n )(b n + ε n ) log(b n + ε n ) 1 12 (b n -a n ) 4 + o((b n -a n ) 4 ) = - 1
|ϕ -ϕ ⋆ | α(ϕ ⋆ )-2 C ′ • (ϕ) |ϕ -ϕ ⋆ | β(ϕ ⋆ )-2 as ϕ → ϕ ⋆ .
Observe that Theorem 5.4 states that if MCP holds, then β(ϕ ⋆ ) must be finite for every ϕ ⋆ . The essence of the proof of Theorem 5.9 is that if the supremum in (81) and the infimum in (80) are equal, finite and attained, then lim sup (ϕ,ω)→(ϕ ⋆ ,0) N (ϕ, ω) is finite. The refinement in Theorem 5.10 shows that if furthermore A(ϕ ⋆ ) = B(ϕ ⋆ ) for every ϕ ⋆ , then lim sup (ϕ,ω)→(ϕ ⋆ ,0) N (ϕ, ω) ≥ 2α(ϕ ⋆ ) + 1 and thus N curv ≥ 2α(ϕ ⋆ ) + 1 for every ϕ ⋆ . One may wonder if these sufficient conditions for MCP are also necessary, and that is what the examples we provided are investigating. Example 5.16 has α(ϕ ⋆ ) = β(ϕ ⋆ ), while the supremum and infimum in (81) and (80) are not attained, and yet the lim sup (ϕ,ω)→(ϕ ⋆ ,0) N (ϕ, ω) is finite. At this point, it is reasonable to ask whether the necessary and sufficient condition for MCP to hold is simply that α(ϕ ⋆ ) = β(ϕ ⋆ ) < +∞. Example 5.17 shows that this is not the case. We constructed an example that has α(ϕ ⋆ ) = β(ϕ ⋆ ) < +∞ with lim sup (ϕ,ω)→(ϕ ⋆ ,0) N (ϕ, ω) = +∞.

In conclusion, α(ϕ ⋆ ) = β(ϕ ⋆ ) < +∞ should be a necessary condition for MCP to hold, and the necessary and sufficient condition for the measure contraction property probably lies in the "oscillation" of the angle correspondence C ′ • near its zero points. There may be two obstacles for achieving this goal. Firstly, one would need to define rigorously a good notion of oscillation. Secondly, this oscillating function would need to be integrated and quantitatively estimated.

6 Failure of the CD(K, N) condition in the sub-Finsler Heisenberg group

In this section we prove Theorem 1.6. Our argument consists in a suitable refinement of the strategy developed in [START_REF] Magnabosco | Failure of the curvature-dimension condition in sub-finsler manifolds[END_REF]Sec. 5.3]. For sake of clarity, we will explicitly adapt the proof of [START_REF] Magnabosco | Failure of the curvature-dimension condition in sub-finsler manifolds[END_REF]Thm. 5.24] to prove Theorem 6.3, as this adaptation consists of several non-trivial improvements. On the contrary, for the fundamental preliminary result (Proposition 6.2) we will simply refer to [START_REF] Magnabosco | Failure of the curvature-dimension condition in sub-finsler manifolds[END_REF], explaining the arguments can be easily adapted to the setting of this section.

We consider the sub-Finsler Heisenberg group H, equipped with a C 1 strongly convex norm • , recalling that in this case the correspondence map C • is single-valued and Lipschitz, thus differentiable almost everywhere. Moreover, according to Proposition 3.5, we know that if p, q ∈ H are such that p ⋆ q -1 / ∈ {x = y = 0}, then there exists unique geodesic joining p and q. Recall the definition of midpoint map: M(p, q) := e 1 2 (γ pq ) , if p ⋆ q -1 / ∈ {x = y = 0}, where γ pq : [0, 1] → H is the unique geodesic joining p and q. Similarly, we define the inverse geodesic map I m (with respect to m ∈ H) as:

I m (q) = p, if M(p, q) = m.
can show that the set {s ∈ I : Φ(s) ∈ D + reg } = Φ -1 (J) ⊂ I has density 1 in 0. In particular, up to taking a smaller interval Ĩ ⊂ I, we can realize (84).

At this point, let s ∈ Ĩ such that Φ(s) ∈ D + reg and consider ρ := ρ G -1 t (α(s)) > 0, where ρ(•) is the positive constant found in Proposition 6.2. For every s ∈ [-ρ, ρ] \ {0}, from Proposition 6.2, we deduce that the inverse geodesic map I e and the midpoint map M are well-defined and C 1 in a neighborhood of G s G -1 t (α(s)) and e, G s G -1 t (α(s)) , respectively. Moreover, we have that det d Gs(G -1 t (α(s))) M(e, •) ≤ 1 2 4 . Observe that, since the left-translations are smooth isometries, the inverse geodesic map I γ(s) is well-defined and C 1 in a neighborhood of γ(s + s), in fact it can be written as I γ(s) (p) = L γ(s) I e L γ(s) -1 (p) , and L γ(s) -1 γ(s + s) = G s G -1 t (α(s)) . Similarly, we can prove that the midpoint map is well-defined and C 1 in a neighborhood of (γ(s), γ(s + s)), with det d γ(s+s) M(γ(s), •) ≤ 1 2 4 . In conclusion, up to restriction and reparametrization, we can find a geodesic η : [0, 1] → H and a set G ⊂ [0, 1] with L 1 (G) := m > 7 8 , cf. (84), such that for every s ∈ G, there exists λ(s) > 0 such that, for every s ∈ [sλ(s), s + λ(s)] ∩ [0, 1] \ {s}, the inverse geodesic map I η(s) and the midpoint map M are well-defined and C 1 in a neighborhood of η(s) and (η(s), η(s)) respectively, and in addition det d η(s) M(η(s), •) ≤ 1 2 4 . Set λ(s) = 0 on the set [0, 1] \ G and consider the set T := (s, t) ∈ [0, 1] 2 : t ∈ [sλ(s), s + λ(s)] .

Observe that, introducing for every ε > 0 the set D ε := {(s, t) ∈ [0, 1] 2 : |t -s| < ε}, we have that

L 2 (T ∩ D ε ) L 2 (D ε ) = L 2 (T ∩ D ε ) 2ε -ε 2 → m > 7 8 , as ε → 0. ( 85
)
On the other hand, we can find δ > 0 such that the set Λ δ := {s ∈ [0, 1] : λ(s) > δ} satisfies L 1 (Λ δ ) > 13 16 . In particular, for every ε < δ sufficiently small we have that

L 2 (s, t) ∈ [0, 1] 2 : s + t 2 ∈ Λ δ ∩ D ε > 3 2 ε > 3 4 L 2 (D ε ). ( 86
)
Therefore, putting together (85) and (86), we can find ε < δ sufficiently small such that

L 2 T ∩ D ε ∩ (s, t) ∈ [0, 1] 2 : s + t 2 ∈ Λ δ > 1 2 L 2 (D ε ).
Then, since the set D ε is symmetric with respect to the diagonal {s = t}, we can find s = t such that (s, t), ( t, s) ∈ T ∩ D ε ∩ (s, t) ∈ [0, 1] 2 : s + t 2 ∈ Λ δ .

In particular, this tells us that:

  CD(K, N ) and the MCP(K, N ) conditions . . . . . . . . . . . . . . . . . . . . 2.2 Convex trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Convex trigonometry associated with a norm . . . . . . . . . . . . . . . . . . . . . 3 The sub-Finsler geometry of the Heisenberg group 3.1 Geodesics in the Heisenberg group . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Jacobian of the exponential map . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The MCP condition in the Heisenberg group . . . . . . . . . . . . . . . . . . . . . .

Figure 1 :

 1 Figure 1: Values of the generalized trigonometric functions cos Ω and sin Ω .

Figure 2 :

 2 Figure 2: Representation of the correspondence θ Ω ← → ψ.

.

  t) 2 |s log |s|| • |t log |t|| ds dt = 1 16 x|x| 3 (1 -4 log |x|) ωt) 2 C ′ • (t) dt = |(ϕ + ω) log |ϕ + ω|| ˆϕ+ω ϕ (ϕ + ωt) 2 |t log |t|| dt = |(ϕ + ω) log |ϕ + ω|| × (ϕ + ω) 2 1 4 x|x|(1 -2 log |x|) ϕ+ω ϕ -2(ϕ + ω) 1 9 |x| 3 (1 -3 log |x|)In the computations above, we have use the notation [F (x)] b a := F (b) -F (a) as well as d dx 1 4 x|x|(1 -2 log |x|) = |x log |x||, d dx 1 9 |x| 3 (1 -3 log |x|) = x|x log |x||, 1 -4 log |x|) = x 2 |x log |x||, for all x ∈ (-1, 1) \ {0}.

  s) 2 ts log s ds dts) 2 st log t ds dt (78) s) 2 ts log t log s ds dt (-a n ) 6 + o((b na n ) 6 ) = 1 72 b 6 n + o(b 6 n ), as n → +∞.The asymptotic above is justified since1 (b na n ) 6ˆbn an ˆbn+εn bn (ts) 2 ts log s ds dt ≤ (b na

12 b 6

 6 n log(b n ) + o(-b 6 n log(b n )),as n → +∞.From (63) and (64), it is also not difficult to see thatR(ϕ n , ω n ) = o(b 6 n ), as well as ∂ ω R(ϕ n , ω n ) = o(-b 5 n log(b n )), as n → +∞. Consequently, we obtain that lim n→+∞ N (ϕ n , ω n ) = lim n→+∞ 1 + -1 12 b 6 n log(b n ) + o(-b 6 n log(b n )) 1 72 b 6 n + o(b 6 n ) = +∞.This shows that the metric measure space (H, d, L 3 ) does not satisfy the MCP(0, N ) for any N ∈ (1, ∞), according to Proposition 3.23.Let us summarise the findings of this section. For ϕ ⋆ ∈ R/2π Ω• Z, we introduce the following constantsα(ϕ ⋆ ) := inf α ≥ 2 : A(ϕ ⋆ ) := sup δ>0 ess inf |ϕ-ϕ ⋆ |<δ C ′ • (ϕ) |ϕϕ ⋆ | α-2 > 0 . (80) and β(ϕ ⋆ ) := sup β ≥ 2 : B(ϕ ⋆ ) := inf δ>0 ess sup |ϕ-ϕ ⋆ |<δ C ′ • (ϕ) |ϕϕ ⋆ | β-2 < +∞ (81)

  •• = Ω and this allows to prove the following symmetry property for the correspondence just defined.

	Proposition 2.11. Let Ω ⊂ R 2 be a convex and compact set, with O ∈ Int(Ω). Given two angles
	θ, ϕ ∈ R, θ	Ω ← → ϕ if and only if ϕ	Ω • ←→ θ. Moreover, the following analogous of the Pythagorean
	equality holds:

  where • * is the dual norm.Definition 2.13 (Strictly and strongly convex norm). We say that • is strictly convex if the function f Ω is strictly convex. Similarly, we say that • is strongly convex if f Ω is strongly convex with respect to • .

Note that, according to [Cio90, Prop. 1.6], • is strictly convex if and only if the associated unit ball Ω = B • 1 (0) is a strictly convex set. Whereas, [Cio90, Prop. 2.11] implies that • is strongly convex if and only if the associated unit ball Ω is a uniformly convex set. We recall below a well-known result on the relation between a norm • and its dual norm • * , cf. [AP95, Prop. 2.6].

  and only if C • is single valued at every angle and continuous, (iii) • is a strongly convex norm if and only if C • is Lipschitz continuous. Proof. (i) According to Proposition 2.14, the norm • is C 1 if and only if the dual norm • * is strictly convex. The equivalence between strict convexity of the reference set (in this case Ω • ) and strict monotonicity of the angle correspondence has been already observed in [Lok19, Sec. 3]. (ii) According to Proposition 2.14, the norm • is strictly convex if and only if the dual norm • * is C 1 . This is equivalent to asking that d • * is continuous. The thesis follows from Lemma 2.15 and Lemma 2.8.

Acknowledgments

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No. 945655). M.M. acknowledges support from the Royal Society through the Newton International Fellowship (award number: NIF\R1\231659). T.R. acknowledges support from the DFG through the collaborative research centre "The mathematics of emerging effects" (CRC 1060, Project-ID 211504053) and

Consider a sequence {(ϕ n , ω n )} n∈N ⊂ U of positive numbers such that ϕ n , ϕ n + ω n ∈ D 1 and (ϕ n , ω n ) → (0, 0) as n → +∞. Assume also, without loss of generality, that the ratio s n := ω n /ϕ n converges to s ∈ [-∞, +∞] as n → +∞.

and similarly

Note that the function P (s) attains 0 only at s = 0, the map s → s∂ s P (s)/P (s) is continuous, and the following limits can be easily established:

The remainder terms R(ϕ n , ω n ) and ∂ ω R(ϕ n , ω n ) (cf. (54) and (55)) can be dealt with in the same way as in the proofs of Theorems 5.9 and 5.10 and they are of higher orders. Therefore, we have that

Similarly, if ω n /ϕ n → ±∞, then for any a > 0, it holds that

In any case, the limit superior of the map (ϕ, ω) → ω∂ ω J R (ϕ, ω)/J R (ϕ, ω) as (ϕ, ω) → (0, 0) remains bounded. Therefore, the Heisenberg group (H, d, L 3 ) satisfies MCP(0, N ) for some N ∈ (1, +∞) according to Proposition 3.23.

In Lemma 3.15, we have identified a L 1 -positive set D + 1 where C • is differentiable with positive derivative. In [START_REF] Magnabosco | Failure of the curvature-dimension condition in sub-finsler manifolds[END_REF], this property played a fundamental role in deducing Jacobian estimates for the exponential map. In particular, with the same strategy we can prove the following proposition, which is the analogous of [MR23b, Prop. 5.23].

Remark 6.1. Recall Definition 3.3 and observe that

In particular, the map G t for t ∈ [-1, 0] has the same properties of G -t , cf. Proposition 3.5 and Proposition 3.11. Proposition 6.2. Let H be the sub-Finsler Heisenberg group, equipped with a C 1 strongly convex norm • . There is a full-measure set D + reg ⊂ D + 1 so that, for every (r, ϕ, ω) ∈ R = U with ϕ ∈ D + reg , there exists a positive constant ρ = ρ(r, ϕ, ω) such that for every t ∈ [-ρ, ρ] \ {0}:

(i) the inverse geodesic map I e is well-defined and C 1 in a neighborhood of G t (r, ϕ, ω);

(ii) the midpoint map M is well-defined and C 1 in a neighborhood of (e, G t (r, ϕ, ω)), moreover

Sketch of the proof. We follow the blueprint of [START_REF] Magnabosco | Failure of the curvature-dimension condition in sub-finsler manifolds[END_REF]. Therein, the map C • is differentiable with positive derivative L 1 -a.e. in R/2π Ω • Z and this allows to deduce the needed estimates for the Jacobian J t (r, ϕ, ω), for every r > 0, ω ∈ (-2π

In this case, according to Lemma 3.15, the set

where C • is differentiable with positive derivative, has only positive L 1 -measure. Nonetheless, the latter is sufficient to prove the estimates of [MR23b, Lem. 5.16-5.19] for every r > 0, ω ∈ (-2π Ω • , 2π Ω • )\{0} and for L 1 -a.e. ϕ ∈ D + 1 . Thus, [MR23b, Cor. 5.20] holds also in this setting for L 1 -a.e. angle in D + 1 . Finally, once the latter result is proven, the proof of the current proposition can be carried out repeating verbatim the one of [MR23b, Prop. 5.23], with D + 1 in place of R/2π Ω • Z.

Theorem 6.3. Let H be the sub-Finsler Heisenberg group, equipped with a C 1 strongly convex norm • and with a smooth measure m. Then, the metric measure space (H, d, m) does not satisfy the Brunn-Minkowski inequality BM(K, N ), for every K ∈ R and N ∈ (1, ∞).

Proof. Take an angle ϕ ∈ D + reg (cf. Proposition 6.2) which is a density point for D + reg . Fix r > 0, ω ∈ (-2π Ω • , 2π Ω • ) \ {0} and, for simplicity, call γ the curve γ (r,ϕ,ω) , i.e.

[0, 1] ∋ s → γ(t) := G t (r, ϕ, ω).

By Remark 3.12, we know that for every t ∈

and observe that, for s sufficiently small, we have α(s) ∈ B t . Therefore, the function

where p i : T * e H → R denotes the projection onto the i-th coordinate, is well-defined and C 1 (being composition of C 1 functions) in an open interval I ⊂ R containing 0. In particular, note that Φ(s) is the "initial angle" for the geodesic joining e and α(s).

We are now going to prove that Φ ′ (0) = 0.

(C)

Consider the map

with non-zero differential at γ(t), the set O := F -1 (ϕ) locally defines a C 1 -surface and its tangent space is T p O = Ker d p F , for every p ∈ O ∩ B t . On the one hand, the tangent space to O is spanned by ∂Gt ∂ω , ∂Gt ∂r and, on the other hand, Φ ′ (0) = d α(0) F ( α(0)). Therefore, (C) is equivalent to showing that

In order to prove (82), we observe that, since the left-translations are isometries and γ is a geodesic,

This implies that p 1 G -1 t α(s) ≡ r for s ∈ I and, as a consequence, reasoning as above with F := p 1 • G -1 t , we deduce that α(0) ∈ span ∂G t ∂ϕ (r, ϕ, ω), ∂G t ∂ω (r, ϕ, ω) .

In conclusion, to prove (C) it is enough to show that

To this aim, we explicitly compute that α(0) = r cos Ω ((ϕ + ωt)

where △ and ♦ denote quantities that we do not need to make explicit. In particular, call M the minor of the (3 × 2)-matrix α(0) ∂G t ∂ω , obtained by erasing the last row and observe that, in order to prove (83), it is sufficient to verify that det M = 0. The explicit computation shows that det M = r 2 ω 2 J R (ϕ, ωt) > 0, cf. Proposition 3.16. This proves (C). Now, we want to find an interval Ĩ ⊂ I (containing 0) such that

Consider the set J := {ψ ∈ Φ(I) : ψ ∈ D + reg } ⊂ Φ(I) and observe that J has density 1 in Φ(0) = ϕ. In fact, ϕ is a density point for D + reg and, as a consequence of (C), Φ(I) contains a neighborhood of ϕ. Then, using once again claim (C), we (i) t ∈ [sλ(s), s + λ(s)] and s ∈ [ tλ( t), t + λ( t)];

(ii) | t -s| < ε < δ;

(iii) s+ t 2 ∈ Λ δ . Now, on the one hand, (i) ensures that the midpoint map M is well-defined and C 1 in a neighborhood of (η(s), η( t)) with det d η( t) M(η(s), and, by the very definition of λ(•), we obtain the claimed regularity of the inverse geodesic map. Once we have these properties, we can repeat the same strategy used in the second part of the proof of [START_REF] Magnabosco | Failure of the curvature-dimension condition in sub-finsler manifolds[END_REF]Thm. 4.26] and contradict the Brunn-Minkowski inequality BM(K, N ) for every K ∈ R and every N ∈ (1, ∞).