Colin Petitjean
email: colin.petitjean@univ-fcomte.fr

LIPSCHITZ-FREE SPACES AND SCHUR PROPERTIES

Keywords: 2010 Mathematics Subject Classification. Primary 46B10, 46B28, Secondary 46B20 Lipschitz-free spaces, Schur property, Compact metric spaces, Proper metric spaces, Quasi-Banach spaces, Approximation property

In this paper we study ℓ 1 -like properties for some Lipschitz-free spaces. The main result states that, under some natural conditions, the Lipschitz-free space over a proper metric space linearly embeds into an ℓ 1 -sum of finite dimensional subspaces of itself. We also give a sufficient condition for a Lipschitz-free space to have the Schur property, the 1-Schur property and the 1-strong Schur property respectively. We finish by studying those properties on a new family of examples, namely the Lipschitz-free spaces over metric spaces originating from p-Banach spaces, for p in (0, 1).

Introduction

Let M be a pointed metric space and fix 0 ∈ M a distinguished origin of M. Let us denote Lip 0 (M) the space of all real valued Lipschitz functions f defined on M and verifying f (0) = 0. This space equipped with the Lipschitz norm • L (best Lipschitz constant) is a Banach space. For x ∈ M, define the evaluation functional δ M (x) ∈ Lip 0 (M) * by δ M (x), f = f (x) for every f ∈ Lip 0 (M). And now we let the Lipschitz-free space over M, denoted F (M), be the norm closed linear span in Lip 0 (M) * of {δ M (x) : x ∈ M}. The map δ M : x ∈ M → δ M (x) ∈ F (M) is readily seen to be an isometry. Moreover the space F (M) enjoys a nice factorization property. For any Banach space X, for any Lipschitz function f : M → X, there exists a unique linear operator f : F (M) → X with f = f L and such that the following diagram commutes

M f / / _ δ M X F (M) f < < ① ① ① ① ① ① ① ① ①
Thus the map f ∈ Lip 0 (M, X) → f ∈ L(F (M), X) is an onto linear isometry. As a direct consequence, F (M) is an isometric predual of Lip 0 (M).

After the seminal paper [START_REF] Godefroy | Lipschitz-free Banach spaces[END_REF] of Godefroy and Kalton, Lispchitz free spaces have become an object of interest for many authors (see for instance [START_REF] Cúth | On the structure of Lipschitz-free spaces[END_REF][START_REF] Dalet | Characterization of metric spaces whose free space is isometric to ℓ 1[END_REF][START_REF] Godard | Tree metrics and their Lipschitz-free spaces[END_REF][START_REF] Godefroy | Free Banach spaces and the approximation properties[END_REF][START_REF] Hájek | Approximation and Schur properties for Lipschitz free spaces over compact metric spaces[END_REF][START_REF] Hájek | On Schauder bases in Lipschitz-free spaces[END_REF][START_REF]Spaces of Lipschitz and Hölder functions and their applications[END_REF]). Indeed, the fundamental factorization property cited above transforms in a particular way a nonlinear problem into a linear one. This creates links between some old open problems in the geometry of Banach spaces and some open problems about Lipschitz-free spaces (see the open problems section in [START_REF] Godefroy | A survey on Lipschitz-free Banach spaces[END_REF]).

In this note we focus mostly on ℓ 1 -like properties like the Schur property or some stronger properties. In [START_REF]Spaces of Lipschitz and Hölder functions and their applications[END_REF], Kalton proved that if (M, d) is a metric space and ω is a nontrivial gauge then the space F (M, ω • d) has the Schur property. In [START_REF] Hájek | Approximation and Schur properties for Lipschitz free spaces over compact metric spaces[END_REF] Hájek, Lancien and Pernecká proved that the Lipschitz-free space over a proper countable metric space has the Schur property.

Here we give a condition on M which ensures that F (M) has the Schur property, and unifies the two above mentioned results. Furthermore, assuming that M is proper (every closed ball is compact) and F (M) has the metric approximation property, we are able to provide more information about the"ℓ 1 -structure" of F (M). More precisely, we have the following theorem which is our first main result.

Theorem 1.1. Let M be a proper metric space such that S 0 (M) separates points uniformly (see Definition 2.1) and such that F (M) has (MAP). Then for any ε > 0, there exists a sequence

(E n ) n of finite-dimensional subspaces of F (M) such that F (M) is (1 + ε)-isomorphic to a subspace of ( ⊕ n E n ) ℓ 1 .
We now describe the content of this paper. In Section 2, we start by introducing some useful tools such as the little Lipschitz space over a metric space M (denoted lip 0 (M)) and the uniform separation of points. Next, generalizing the proof of Theorem 4.6 in [START_REF]Spaces of Lipschitz and Hölder functions and their applications[END_REF], we show that F (M) has the Schur property whenever lip 0 (M) is 1-norming. Next we move to the proof of Theorem 2.7 and we also show that it is optimal in some sense. Then we show that some quantitative versions of the Schur property are inherited by Lipschitz-free spaces satisfying some of the assumptions of Theorem 2.7.

In Section 3, we introduce a new family of metric spaces where some results of Section 2 apply. This family consists of some metric spaces originating from p-Banach spaces (p ∈ (0, 1)). We first focus on the metric space M n p originating from ℓ n p for which we show that F (M n p ) satisfies the hypothesis of Theorem 2.7. Then we extend this result for arbitrary finite dimensional p-Banach spaces (Corollary 3.5). As a consequence we obtain our second main result. Theorem 1.2. Let p be in (0, 1) and let (X, • ) be a p-Banach space which admits a finite dimensional decomposition. Then F (X, • p ) has the Schur property.

Finally in Section 4 we give some open problems and make some related comments.

Schur properties and Lipschitz-free spaces

In this note, we only consider real Banach spaces and we write X ≡ Y to say that the Banach spaces X and Y are linearly isometric.

2.1. The little Lipschitz space and the uniform separation of points. Definition 2.1. Let M be a metric space. We define the two following closed subspaces of Lip 0 (M) (the second one differs from the first one when M is unbounded).

lip 0 (M) := f ∈ Lip 0 (M) : lim ε→0 sup 0<d(x,y)<ε |f (x) -f (y)| d(x, y) = 0 , S 0 (M) :=    f ∈ lip 0 (M) : lim r→∞ sup x or y / ∈B(0,r) x =y |f (x) -f (y)| d(x, y) = 0    .
The first space lip 0 (M) is called the little Lipschitz space over M. This terminology first appeared in [START_REF] Weaver | Lipschitz algebras[END_REF] where they were considered on compact metric spaces. Definition 2.2. We will say that a subspace S ⊂ Lip 0 (M) separates points uniformly if there exists a constant c ≥ 1 such that for every x, y ∈ M there is f ∈ S satisfying ||f || L ≤ c and f (x) -f (y) = d(x, y). We then say that a subspace Z of X * , where X is a Banach space, is C-norming (with C ≥ 1) if for every x in X,

x ≤ C sup z * ∈B Z |z * (x)|.
It is well known that lip 0 (M) or S 0 (M) are C-norming for some C ≥ 1 if and only if they separate points uniformly (see Proposition 3.4 in [START_REF]Spaces of Lipschitz and Hölder functions and their applications[END_REF]).

Weaver showed in [START_REF] Weaver | Lipschitz algebras[END_REF] (Theorem 3.3.3) that if M is a compact metric space then lip 0 (M) separates points uniformly if and only if it is an isometric predual of F (M). More generally, Dalet showed in [START_REF] Dalet | Free spaces over some proper metric spaces[END_REF] that a similar result holds for proper metric spaces. We state here this result for future reference.

Theorem 2.3 (Dalet). Let M be a proper metric space. Then S 0 (M) separates points uniformly if and only if it is an isometric predual of F (M).

The following proposition will be useful in Section 3. It provides conditions on M to ensure that lip 0 (M) 1-norming.

Proposition 2.4. Let M be a metric space. Assume that for every x = y ∈ M and ε > 0, there exist N ⊆ M and a

(1 + ε)-Lipschitz map T : M → N such that lip 0 (N) is 1-norming for F (N), d(T x, x) ≤ ε and d(T y, y) ≤ ε. Then lip 0 (M) is 1-norming.
Proof. Let x = y ∈ M and ε > 0. By our assumptions there exist N ⊆ M and a (1+ε)-Lipschitz map T : M → N such that lip 0 (N) is 1-norming, d(T x, x) ≤ ε and d(T y, y) ≤ ε. Since lip 0 (N) is 1-norming there exists f ∈ lip 0 (N) verifying f L ≤ 1 + ε and |f (T x) -f (T y)| = d(T x, T y). Now we define g = f • T on M. By composition g is (1 + ε) 2 -Lipschitz and g ∈ lip 0 (M). Then a direct computation shows that g does the work.

|g(x) -g(y)| = |f (T x) -f (T y)| = d(T x, T y) ≥ d(x, y) -d(x, T x) -d(y, T y) ≥ d(x, y) -2ε.
This ends the proof.

2.2. The Schur property. We now turn to the study of the classical Schur property. We first recall its definition.

Definition 2.5. Let X be a Banach space. We say that X has the Schur property if every weakly null sequence (x n ) n∈N in X is also • -convergent to 0.

According to [START_REF]Spaces of Lipschitz and Hölder functions and their applications[END_REF], by a gauge we mean a continuous, subadditive and increasing function ω : [0, ∞) -→ [0, ∞) verifying ω(0) = 0 and ω(t) ≥ t for every t ∈ [0, 1]. We say that a gauge ω is non-trivial whenever lim t→0 ω(t) t = ∞. In [START_REF]Spaces of Lipschitz and Hölder functions and their applications[END_REF] (Theorem 4.6), Kalton proved that if (M, d) is a metric space and ω is a nontrivial gauge then the space F (M, ω • d) has the Schur property. A careful reading of his proof reveals that the key ingredient is actually the fact that lip 0 (M, ω •d) is always 1-norming (Proposition 3.5 in [START_REF]Spaces of Lipschitz and Hölder functions and their applications[END_REF]). This leads us to the following result. We include it here for completeness even though the proof is very similar to Kalton's original argument.

Proposition 2.6. Let M be a metric space such that lip 0 (M) is 1-norming. Then the space F (M) has the Schur property.

Proof. According to Proposition 4.3 in [START_REF]Spaces of Lipschitz and Hölder functions and their applications[END_REF], for every ε > 0, F (M) is (1 + ε)-isomorphic to a subspace of ( n∈Z F (M k )) ℓ 1 where M k denotes the ball B(0, 2 k ) ⊆ M centered at 0 and of radius 2 k . Moreover the Schur property is stable under ℓ 1 -sums, under isomorphism and passing to subspaces. So it suffices to prove the result under the assumption that M has finite radius.

Let us consider (γ n ) n a normalized weakly null sequence in F (M). We will show that

∀γ ∈ F (M), lim inf n→+∞ γ + γ n ≥ γ + 1 2 , (1)
from which it is easy to deduce that for every ε > 0, (γ n ) n admits a subsequence (2 + ε)equivalent to the ℓ 1 -basis (see the end of the proof of Proposition 4.6 in [START_REF]Spaces of Lipschitz and Hölder functions and their applications[END_REF]). This contradicts the fact that (γ n ) n is weakly null.

Fix ε > 0 and γ ∈ F (M). We can assume that γ is of finite support. Pick f ∈ lip 0 (M) with

f L = 1 and f, γ > γ -ε. Next pick Θ > 0 so that if d(x, y) ≤ Θ then |f (x) -f (y)| < εd(x, y). Choose δ < εΘ 2(1+ε) .
Then by Lemma 4.5 in [START_REF]Spaces of Lipschitz and Hölder functions and their applications[END_REF] we have inf

|E|<∞ sup n dist(γ n , F ([E] δ )) = 0,
where [E] δ = {x ∈ M : d(x, E) ≤ δ}. Thus there exists a finite set E ⊂ M such that E contains the support of γ and such that for each n we can find µ n ∈ F ([E] δ ) with γ n -µ n < ε. Remark that F (E) is a finite dimensional space. Thus lim inf n→+∞ dist(γ n , F (E)) ≥ 1 2 . Then by Hahn-Banach theorem, for every n there exists f n ∈ Lip 0 (M) verifying

f n L ≤ 1 + ε, f n (E) = {0} and lim inf n→+∞ f n , γ n ≥ 1 2 . Now we define g n = (f + f n ) |[E] δ , then g n ∈ Lip 0 ([E] δ
) and we will show that g n L < 1 + ε. We will distinguish two cases to show this last property. First suppose that x and y are such that d(x, y) ≤ Θ, then

|g n (x) -g n (y)| ≤ |f (x) -f (y)| + |f n (x) -f n (y)| ≤ εd(x, y) + d(x, y) = (1 + ε)d(x, y).
Second if x and y are such that d(x, y) > Θ, then there exist u, v ∈ E with d(x, u) ≤ δ and d(y, v) ≤ δ, so that

|g n (x) -g n (y)| ≤ |f (x) -f (y)| + |f n (x)| + |f n (y)| = |f (x) -f (y)| + |f n (x) -f n (u)| + |f n (y) -f n (v)| ≤ d(x, y) + 2(1 + ε)δ ≤ d(x, y) + εΘ ≤ (1 + ε)d(x, y).
We extend those functions g n to M with the same Lipschitz constant and we still denote those extensions g n for convenience. We now estimate the desired quantities.

γ + µ n ≥ 1 1 + ε g n , γ + µ n = 1 1 + ε ( f, γ + f, µ n + f n , γ + f n , µ n ), where (i) f, γ > γ -ε. (ii) lim sup n→∞ | f, µ n | ≤ ε, since (γ n ) n is weakly null and γ n -µ n < ε. (iii) f n , γ = 0, since γ ∈ F (E). (iv) lim inf n→∞ f n , µ n ≥ 1 2 -ε, since lim inf n→∞ f n , γ n > 1 2 . Thus, lim inf n→∞ γ + γ n ≥ 1 1 + ε ( γ + 1 2 -3ε) -ε.
Since ε is arbitrary, this proves (1).

2.3.

Embeddings into ℓ 1 -sums. Before stating the main result of this section we recall a few classical definitions. We say that a Banach space X has the approximation property (AP) if for every ε > 0, for every compact set K ⊂ X, there exists a finite rank operator T ∈ B(X) such that T x-x ≤ ε for every x ∈ K. Let λ ≥ 1, if in the above definition T can always be chosen so that T ≤ λ, then we say that X has the λ-bounded approximation property (λ-(BAP)). When X has the 1-(BAP) we say that X has the metric approximation property (MAP). We can now state and prove our first main result.

Theorem 2.7. Let M be a proper metric space such that S 0 (M) separates points uniformly and such that F (M) has (MAP). Then for any ε > 0, there exists a sequence

(E n ) n of finite-dimensional subspaces of F (M) such that F (M) is (1 + ε)-isomorphic to a subspace of ( ⊕ n E n ) ℓ 1 .
Proof. The proof is based on three results. The first ingredient is the following lemma (Lemma 3.1 in [START_REF] Godefroy | On subspaces of L 1 which embed into l 1[END_REF]):

Lemma 2.8 (Godefroy, Kalton and Li). Let V be a subspace of c 0 (N) with (MAP). Then for any ε > 0, there exists a sequence (E n ) n of finite-dimensional subspaces of V * and a weak * -toweak * continuous linear map T :

V * → ( ⊕ n E n ) ℓ 1 such that for all x * ∈ V * (1 -ε) x * ≤ T x * ≤ (1 + ε) x * .
Next, generalizing a proof of Kalton in the compact case (Theorem 6.6 in [START_REF]Spaces of Lipschitz and Hölder functions and their applications[END_REF]), Dalet has proved the following lemma (Lemma 3.9 in [START_REF] Dalet | Free spaces over some proper metric spaces[END_REF]) Lemma 2.9 (Dalet). Let M be a proper metric space. Then, for every ε > 0, the space S 0 (M)

is (1 + ε)-isomorphic to a subspace Z of c 0 (N).
Finally we need the following two results about (MAP) (see [START_REF] Grothendieck | Produits tensoriels topologiques et espaces nucléaires[END_REF] and Theorem 1.e.15 in [START_REF] Lindenstrauss | Classical Banach spaces[END_REF]).

Theorem 2.10 (Grothendieck). Let X be a Banach space. (G1). If X * has (MAP) then X has (MAP). (G2). If X * is separable and has (AP) then X * has (MAP).

We are ready to prove Theorem 2.7. Let us consider a metric space M satisfying the assumptions of Theorem 2.7 and let us take ε > 0 arbitrary. Fix ε ′ such that (1+ε ′ ) 3 < 1+ε. According to Lemma 2.9, there exists a subspace Z of c 0 (N) such that S 0 (M) is (1 + ε ′ )-isomorphic to Z. Then note that Z also has the metric approximation property. Indeed Z * is (1 + ε ′ )-isomorphic to F (M), so Z * has the (1+ε ′ )-bounded approximation property. Next, using (G2) of Theorem 2.10 we get that Z * has MAP. Then using (G1) of Theorem 2.10 we get that Z also has MAP. Thus we can apply Lemma 2.8 to Z so that there exists a sequence (F n ) n of finite-dimensional subspaces of 3 -isomorphic to E. This completes the proof.

Z * such that Z * is (1 + ε ′ )-isomorphic to a subspace F of ( ⊕ n F n ) ℓ 1 . Now F (M) is (1 + ε ′ )-isomorphic to Z * so there exists a sequence (E n ) n of finite-dimensional subspaces of F (M) such that ( ⊕ n E n ) ℓ 1 is (1 + ε ′ )-isomorphic to ( ⊕ n F n ) ℓ 1 . Then there exists a sub- space E of ( ⊕ n E n ) ℓ 1 which is (1 + ε ′ )-isomorphic to F . It is easy to check that F (M) is (1 + ε ′ )
We now give some examples where Theorem 2.7 applies.

Examples 2.11. The space S 0 (M) separates points uniformly and F (M) has (MAP) in any of the following cases. 1. M proper countable metric space (Theorem 2.1 and Theorem 2.6 in [START_REF] Dalet | Free spaces over some proper metric spaces[END_REF]). 2. M proper ultrametric metric space (Theorem 3.5 and Theorem 3.8 in [START_REF] Dalet | Free spaces over some proper metric spaces[END_REF]). 3. M compact metric space such that there exists a sequence (ε n ) n tending to 0, a real number ρ < 1/2 and finite ε n -separated subsets N n of M which are ρε n -dense in M (Proposition 6 in [START_REF] Godefroy | Free Banach spaces and the approximation properties[END_REF]). For instance the middle-third Cantor set.

Before ending this section, we state a Proposition which says that Theorem 2.7 is optimal in some sense. Indeed, in our following example the dimension of the finite dimensional space E n in Theorem 2.7 has to go to infinity when n goes to infinity. Proposition 2.12. There exists a countable compact metric space K, made of a convergent sequence and its limit, such that F (K) fails to have a cotype. In particular, K satisfies the assumptions of Theorem 2.7 but does not embed isomorphically into ℓ 1 .

Proof. It is well known that c 0 has no nontrivial cotype. Since c 0 is separable, by Godefroy-Kalton lifting theorem (Theorem 3.1 in [START_REF] Godefroy | Lipschitz-free Banach spaces[END_REF]), there is a subspace of F (c 0 ) which is linearly isometric to c 0 . Thus F (c 0 ) has no nontrivial cotype. So for every n ≥ 1, there exist

γ n 1 , • • • , γ n kn ∈ F (c 0 ) such that kn i=1 γ n i n 1 n > n E kn i=1 ε i γ n i n 1 n
, where (ε i ) kn i=1 is an independent sequence of Rademacher random variables. Next we approximate each γ n i by a finitely supported element

µ n i ∈ F (c 0 ) such that kn i=1 µ n i n 1 n > n 2 E kn i=1 ε i µ n i n 1 n
.

Then we define M n = (∪ kn i=1 supp(µ n i )) ∪ {0} ⊂ c 0 which is a finite pointed metric space. Since scaling a metric space does not affect the linear isometric structure of the corresponding Lipschitz-free space, we may and do assume that the diameter of M n is less than 1 2 n+1 . Now we construct the desired compact pointed metric space as follows. Let us define the countable set K := (∪ n≥2 {n} × M n ) ∪ {e}, e being the distinguished point of M. To simplify the notation we write M ′ n for {n} × M n . Then we define a metric d on K such that d(x, e) = 1

2 n if x ∈ M ′ n , d(x, y) = d Mn (x n , y n ) if x = (n, x n ), y = (n, y n ) ∈ M ′ n and d(x, y) = d(x, e)+d(y, e) if x ∈ M ′ n , y ∈ M ′ m with n = m.
Of course, with this metric K is compact since it is a convergent sequence. Moreover the fact that d(x, y) = d(x, e) + d(y, e) for x ∈ M ′ n , y ∈ M ′ m with n = m readily implies that F (K) ≡ ( F (M n ∪ {e})) ℓ 1 (see Proposition 5.1 in [START_REF] Kaufmann | Products of Lipschitz-free spaces and applications[END_REF] for instance). By construction of M n , ( F (M n )) ℓ 1 has no nontrivial cotype. Thus F (K) also has no cotype. Therefore, F (K) cannot embed into ℓ 1 . But since it is a compact countable metric space, lip 0 (K) separates points uniformly and F (K) has (MAP) (Theorem 2.1 and 2.6 in [START_REF] Dalet | Free spaces over some proper metric spaces[END_REF]).

2.4.

Quantitative versions of the Schur property. It is well known that the Schur property is equivalent to the following condition: for every δ > 0, every δ-separated sequence (x n ) n∈N in the unit ball of X contains a subsequence that is equivalent to the unit vector basis of ℓ 1 . That is there exists a constant K > 0 (which may depend on the sequence considered) and a subsequence (

x n k ) k∈N such that n i=1 |a i | ≥ n i=1 a i x n i ≥ 1 K n i=1 |a i | , for every (a i ) n i=1 ∈ R n .
In this case, we say that the sequence (x n k ) k∈N is K-equivalent to the unit vector basis of ℓ 1 . This equivalence can be easily deduced using Rosenthal's ℓ 1 theorem (see [START_REF] Wnuk | The strong Schur property in Banach lattices[END_REF]). This last fact leads us to define the following quantitative version of the Schur property which has been introduced for the first time by Johnson and Odell in [START_REF] Johnson | Subspaces of L p which embed into l p[END_REF]. We also refer to [START_REF] Godefroy | On subspaces of L 1 which embed into l 1[END_REF] for the 1-strong Schur property.

Definition 2.13. Let X be a Banach space. We say that X has the strong Schur property if there exists a constant K > 0 such that, for every δ > 0, any δ-separated sequence (x n ) n∈N in the unit ball of X contains a subsequence that is K δ -equivalent to the unit vector basis of ℓ 1 . If in this definition, K can be chosen so that K = 2 + ε for every ε > 0, then we say that X has the 1-strong Schur property.

It is clear with the above characterization of the Schur property that the strong Schur property implies the Schur property. It is known that the Schur property is strictly weaker than the strong Schur property (see [START_REF] Wnuk | The strong Schur property in Banach lattices[END_REF] or [START_REF] Kalenda | On a difference between quantitative weak sequential completeness and the quantitative Schur property[END_REF] for example). We refer the reader to [START_REF] Kalton | On subspaces of c 0 and extension of operators into C(K)-spaces[END_REF] (Proposition 2.1) for some equivalent formulations of the strong Schur property. Examples 2.14. 1. In [START_REF] Knaust | On c 0 sequences in Banach spaces[END_REF] (Proposition 4.1), Knaust and Odell proved that if X has the property (S) and does not contain any isomorphic copy of ℓ 1 , then X * has the strong Schur property. In particular ℓ 1 and all its subspaces have the strong Schur property. A Banach space has the property (S) if every normalized weakly null sequence contains a subsequence equivalent to the unit vector basis of c 0 . This is known to be equivalent to the hereditary Dunford-Pettis property (Proposition 2 in [2]) 2. In [START_REF] Godefroy | On subspaces of L 1 which embed into l 1[END_REF] (Lemma 3.4), Godefroy, Kalton and Li proved that a subspace of L 1 has the strong Schur property if and only if its unit ball is relatively compact in the topology of convergence in measure.

We now give the second quantitative version of the Schur property which has been introduced more recently by Kalenda and Spurný in [START_REF] Kalenda | On a difference between quantitative weak sequential completeness and the quantitative Schur property[END_REF]. Definition 2.15. Let X be a Banach space, and let (x n ) n∈N be a bounded sequence in X. We write clust X * * (x n ) for the set of all weak * cluster points of (x n ) n∈N in X * * . Then we define the two following moduli:

δ(x n ) := diam{clust X * * (x n )} ca(x n ) := inf n∈N diam{x k ; k ≥ n}.
The first moduli measures how far is the sequence from being weakly Cauchy and the second one measures how far is the sequence from being • -Cauchy. We then say that X has the C-Schur property if for every bounded sequence

(x n ) n∈N in X: ca(x n ) ≤ C δ(x n )
In [START_REF] Kalenda | On a difference between quantitative weak sequential completeness and the quantitative Schur property[END_REF] the authors proved that the 1-Schur property implies the 1-strong Schur property, and that the 1-strong Schur property implies the 5-Schur property. To the best of our knowledge, the question whether the 1-strong Schur property implies the 1-Schur property is open.

In [START_REF]On quantitative Schur and Dunford-Pettis properties[END_REF] it is proved (Theorem 1.1) that if X is a subspace of c 0 (Γ), then X * has the 1-Schur property. So we easily deduce the following proposition. Proposition 2.16. Let M be a proper metric space such that S 0 (M) separates points uniformly. Then F (M) has the 1-Schur property.

Proof. Fix ε > 0. We use Lemma 2.9 to find a subspace Z of c 0 (N) which is (1 + ε)-isomorphic to S 0 (M). Now, according to Theorem 1.1 in [START_REF]On quantitative Schur and Dunford-Pettis properties[END_REF], Z * has the 1-Schur property. Since S 0 (M) separates points uniformly, we have that S 0 (M) * ≡ F (M) (Theorem 2.3). Thus F (M) is (1 + ε)-isomorphic to Z * . Therefore F (M) has the (1 + ε)-Schur property. Since ε is arbitrary, F (M) has the 1-Schur property.

Examples 2.17. It is clear that metric spaces of the Examples 2.11 satisfy the assumptions of Proposition 2.16. Moreover, the following family of examples also satisfies those last assumptions (see Proposition 2.5 in [START_REF] García-Lirola | On the structure of spaces of vector-valued Lipschitz functions[END_REF]). Let M be a proper metric space and ω be a nontrivial gauge. Then (M, ω • d) is a proper metric space such that S 0 (M, ω • d) separates points uniformly. Now this result leads us to wonder if F (M, ω • d) has MAP for M and ω as above (see Question 4.2).

Lipschitz-free spaces over metric spaces originating from p-Banach spaces

In this section we study Lipschitz-free spaces over a new family of metric spaces, namely metric spaces originating from p-Banach spaces. We prove that results of the previous Section 2 can be applied to some spaces in this new family of examples.

Let X be a real vector space and p in (0, 1). We say that a map N: X → [0, ∞) is psubadditive if N(x + y) p ≤ N(x) p + N(y) p for every x and y in X. Then a homogeneous and p-subadditive map • : X → [0, ∞) is called a p-norm if x = 0 if and only if x = 0. Moreover the map (x, y) ∈ X 2 → x -y p defines a metric on X. If X is complete for this metrizable topology, we say that X is a p-Banach space. Note that a p-norm is actually a quasi-norm. That is there is a constant C ≥ 1 such that for every x, y ∈ X: x + y ≤ C( x + y ). Moreover an important theorem of Aoki and Rolewicz implies that every quasi-normed space can be renormed to be a p-normed space for some p in (0, 1). For background on quasi-Banach spaces and p-Banach spaces we refer the reader to [START_REF]Quasi-Banach spaces[END_REF][START_REF] Kalton | An F -space sampler[END_REF].

We fix p in (0, 1) and consider (X, • ) a p-Banach space. We denote M p = (X, d p ) the metric space where the metric is the p-norm of X to the power p: d p (x, y) = x -y p . Now M p being a metric space, we can study its Lipschitz-free space. At this point we would like to mention the paper [START_REF] Albiac | Lipschitz structure of quasi-Banach spaces[END_REF] of Albiac and Kalton in which they define and study Lipschitz-free spaces over p-Banach spaces (and not over metric spaces originating from p-Banach spaces, as we do here). They show among other things that the analogue of Godefroy-Kalton lifting theorem (Theorem 3.1 in [START_REF] Godefroy | Lipschitz-free Banach spaces[END_REF]) is false for p-Banach spaces.

As mentioned before, we know that lip 0 (M, ω • d) is always 1-norming when ω is a nontrivial gauge (Proposition 3.5 in [START_REF]Spaces of Lipschitz and Hölder functions and their applications[END_REF]). But in our case we consider a quasi-norm composed with the nontrivial gauge ω(t) = t p . Thus, we can expect to have the same result. We will see that we need to use more arguments to overpass the difference between a norm and a p-norm, that is the absence of the triangle inequality for the p-norm. However, techniques that are employed here are inspired by Kalton's ideas in [START_REF]Spaces of Lipschitz and Hölder functions and their applications[END_REF].

As usual, • 1 denotes the ℓ 1 -norm on R n . We also denote • p the p-norm on R n defined by

x p = ( n i=1 |x i | p ) 1 p for every x = (x i ) n i=1 ∈ R n .
We begin with a very basic lemma. Lemma 3.1. Let p ∈ (0, 1) and n ∈ N. Then we have the following inequalities

∀x ∈ R n , x 1 ≤ x p ≤ n 1-p p x 1 .
Proof. The inequality x 1 ≤ x p is obvious and a simple application of Hölder's inequality gives

x p ≤ n 1-p p x 1 .
From now on we write M n p for (R n , d p ) = (R n , • p p ). In order to prove our first result about the structure of F (M n p ), we need the following technical lemma. Lemma 3.2. Let R ∈ (0, ∞), p ∈ (0, 1) and n ∈ N. Then, there exists a Lipschitz function ϕ: M n p → M n p such that ϕ is the identity map on B(0, R), is null on M n p \B(0, 2R) and ϕ is n 2-p -Lipschitz.

Proof. Let us define A = B(0, R) ∪ (M n p \B(0, 2R)) ⊂ M n p (balls are considered for d p ) and φ: (A, d p ) → M n p such that φ is the identity on B(0, R) and is null on M n p \B(0, 2R). It is easy to check that φ is 1-Lipschitz. We now write φ

= (φ 1 , • • • , φ n ). Then for every k φ k : (A, d p ) → (R, | • | p ) is 1-Lipschitz. Thus φ k : (A, • p ) → (R, | • |) is also 1-Lipschitz (
k : (R n , • 1 ) → (R, | • |) which is n 1-p p
Lipschitz and ϕ k|A = φ k . Now the left hand side of the inequality in Lemma 3.

1 implies that ϕ k : (ℓ n p , • p ) → (R, | • |) is n 1-p p - Lipschitz. So ϕ k : M n p → (R, | • | p ) is n 1-p Lipschitz. It follows easily that ϕ = (ϕ 1 , • • • , ϕ n ): M n p → M n p is n 2-p
-Lipschitz and verifies the desired properties.

We are now able to prove the following result.

Proposition 3.3. Let p ∈ (0, 1) and n ∈ N. We recall our notation

(M n p , d p ) = (R n , • p p ). Then, the space F (M n p ) is isometric to S 0 (M n p ) * . In particular S 0 (M n p ) is 1-norming.
Proof. In order to prove this result, we will first prove that S 0 (M n p ) is C n -norming for some C n > 0. And then we will deduce the desired result using Theorem 2.3.

For every m ∈ N and t ≥ 0, we define the following function ω m (t) = min{t p , mt} which is continuous, non-decreasing and subadditive. Note that lim m→+∞ ω m (t) = t p . Let x = y ∈ M n p . Since (ℓ n 1 ) * ≡ ℓ n ∞ , by Hahn-Banach theorem there exists x * ∈ ℓ n ∞ such that x * ∞ = 1 and x * , x -y = x -y 1 . According to Lemma 3.1 this gives x * , x -y ≥ n p-1 p

x -y p . From now on we denote F := n 1-p p x * and we see F as an element of (l n p

) * ≡ l n ∞ of norm F (l n p ) * ≤ n 1-p p which satisfies (2) |F (x) -F (y)| ≥ x -y p
Let us consider R > 2 max( x p p , y p p ) and ϕ : M n p → M n p given by Lemma 3.2 (we denote C its Lipschitz constant). Of course we can see ϕ as a C 1 p -Lipschitz function from ℓ n p to ℓ n p . We then consider f m defined on M n p by

f m (z) = ω m (|F (ϕ(z)) -F (y)|) -ω m (|F (y)|).
Let us prove that those functions f m belong to S 0 (M n p ) and do the job. For

z = z ′ ∈ M n p we compute |f m (z) -f m (z ′ )| = | ω m (|F (ϕ(z)) -F (y)|) -ω m (|F (ϕ(z ′ )) -F (y)|) | ≤ ω m (|F (ϕ(z)) -F (ϕ(z ′ ))|) = ω m (|F (ϕ(z) -ϕ(z ′ ))|).
By its definition ω m (t) ≤ t p . So we have

|f m (z) -f m (z ′ )| ≤ |F (ϕ(z) -ϕ(z ′ ))| p ≤ n 1-p d p (ϕ(z), ϕ(z ′ )) ≤ Cn 1-p d p (z, z ′ ). Thus, f m is d p -Lipschitz with f m L ≤ Cn 1-p . Now since ω m (t) ≤ mt we get |f m (z) -f m (z ′ )| ≤ m|F (ϕ(z) -ϕ(z ′ ))| ≤ mn 1-p p ϕ(z) -ϕ(z ′ ) p ≤ C 1 p mn 1-p p z -z ′ p ≤ (C 1 p mn 1-p p z -z ′ 1-p p ) d p (z, z ′ ). Since 1 -p > 0, z -z ′ 1-p p
and thus the Lipschitz constant of f m can be as small as we want for small distances. This provides the fact that f m ∈ lip 0 (M n p ). It remains to prove that f m satisfies the flatness condition at infinity to get f m ∈ S 0 (M n p ). To this end, fix ε > 0 and pick k > 2 such that 2Cn 1-p (k -2) ≤ ε. Now let z and z ′ be in M, and let us discuss by cases:

(i) If z ∈ B(0, kR) and z ′ ∈ B(0, 2R), then |f m (z) -f m (z ′ )| = 0 < ε. (ii) If z ∈ B(0, kR) and z ′ ∈ B(0, 2R), then |f m (z) -f m (z ′ )| d p (z, z ′ ) ≤ |F (ϕ(z ′ ))| p (k -2)R ≤ n 1-p ϕ(z ′ ) p p (k -2)R ≤ Cn 1-p (2R) (k -2)R ≤ ε.
Since ε is arbitrary, this proves that f m ∈ S 0 (M). To finish the first part of the proof just notice now that using the inequality (2) and the fact that lim m→+∞ ω m (t) = t p , we get

|f m (x) -f m (y)| = ω m (|F (x) -F (y)|) ≥ ω m ( x -y p ) -→ m→+∞ d p (x, y).
Thus S 0 (M n p ) is Cn 1-p -norming. We are now moving to the duality argument. Remark that M n p is a proper metric space, so using Theorem 2.3, we have that S 0 (M n p ) * ≡ F (M n p ). Thus, obviously S 0 (M n p ) is 1-norming. Of course this last result still holds for every metric space originating from a p-Banach space X p of finite dimension. As announced, the assumptions of Theorem 2.7 are satisfied for a metric space originating from a p-Banach space of finite dimension. Corollary 3.5. Let p ∈ (0, 1). Consider (M p , d p ) = (X, • p ) where (X, • ) is a p-Banach space of finite dimension. Then, there exists a sequence

(E n ) n of finite-dimensional subspaces of F (M p ) such that F (M p ) is (1 + ε)-isomorphic to a subspace of ( ⊕ n E n ) ℓ 1 .
Proof. The aim is to show that all assumptions of Theorem 2.7 are satisfied for F (M p ). According to Corollary 3.4, S 0 (M p ) is 1-norming. Now it is proved in [START_REF] Lancien | Approximation properties and Schauder decompositions in Lipschitz-free spaces[END_REF] (Corollary 2.2) that if M is a doubling metric space (that is there exists D(M) ≥ 1 such that any ball B(x, R) can be covered by D(M) open balls of radius R/2) then F (M) has the BAP. Since M p ⊂ R n , we get that M p is doubling. Thus in our case F (M p ) has the BAP. Since it is a dual space, we get from Theorem 2.10 that F (M p ) actually has the (MAP). Thus all the assumptions of Theorem 2.7 are satisfied.

We now turn to the study of the structure of F (M p ) with more general assumptions on M p . In particular we now pass to infinite dimensional spaces and the aim is to explore the behavior of F (M p ) regarding properties such as the (MAP) and the Schur property. To do so, we will assume that X is a p-Banach space which admits a Finite Dimensional Decomposition (shortened in FDD). In particular a space which admits a Schauder basis such as ℓ p satisfies this assumption. We start with the study of the Schur property. Using our Proposition 2.4 we manage to prove the following result. Theorem 3.6. Let p in (0, 1) and let (X, • ) be a p-Banach space which admits an FDD. Then F (X, • p ) has the Schur property.

Proof. First of all, note that we can assume that X admits a monotone FDD. Indeed, it is classical that we can define an equivalent p-norm |||•||| on X such that the finite dimensional decomposition is monotone for (X, |||•|||) (see Theorem 1.8 in [START_REF] Kalton | An F -space sampler[END_REF] for instance). Now from the fact that (X, |||•|||) and (X, • ) are isomorphic we deduce that (X, |||•||| p ) and (X, • p ) are Lipschitz equivalent. Thus, by a routine argument for Lipschitz-free spaces, this implies that F (X, |||•||| p ) and F (X, • p ) are isomorphic. Since the Schur property is stable under isomorphism, F (X, |||•||| p ) has the Schur property if and only if F (X, • p ) has the Schur property. So from now on we assume that the FDD is monotone.

The aim is to apply Proposition 2.4. We denote again (M p , d p ) = (X, • p ). Since X admits a monotone FDD, there exists a sequence (X k ) k∈N of finite dimensional subspaces of X such that every x ∈ X admits a unique representation of the form x = ∞ k=1 x k with x k ∈ X k . If we denote P n the projections from X to X n defined by P n (x) = n k=1 x k then sup n P n = 1. Notice that those projections are actually 1-Lipschitz from M p to M p,n where M p,n = (X n , d p ).

Fix x = y ∈ M p and ε > 0. We can write x = ∞ k=1 x k , y = ∞ k=1 y k with x k , y k ∈ X k for every k. Now fix N ∈ N such that d p (x, P N (x)) < ε and d p (y, P N (y)) < ε. Since each X k is of finite dimension, the space ( N k=1 X k , • ) is of finite dimension and thus by Corollary 3.4, S 0 (A) is 1 norming where A = ( N k=1 M p,k , d p ). So in particular lip 0 (A) is 1-norming. Thus, according to Proposition 2.4, lip 0 (M p ) is 1-norming and so F (M p ) has the Schur property by Proposition 2.6. Remark 3.7. Notice that our Theorem 3.6 is not a special case of Theorem 4.6 in [START_REF]Spaces of Lipschitz and Hölder functions and their applications[END_REF]. Indeed, in general we cannot write the distance • p originating from a p-norm as the composition of a gauge and another distance. Let us prove this for instance in M 2 p = (R 2 , • p p ). We argue by contradiction and so we assume that there exists ω a nontrivial gauge and d a distance on R 2 such that x -y p p = ω(d(x, y)) for every x and y in R 2 . Now we consider the points x = (t x , 0), y = (0, t y ). Straightforward computations shows that x -y p p = |t x | p + |t y | p = x p p + y p p . Since d is a distance and ω is a gauge we have

|t x | p + |t y | p = ω(d(x, y)) ≤ ω(d(x, 0) + d(y, 0)) ≤ ω(d(x, 0)) + ω(d(y, 0)) = |t x | p + |t y | p .
Thus ω(d(x, 0) + d(y, 0)) = ω(d(x, 0)) + ω(d(y, 0)). From where we deduce that ω is additive, and so is such that ω(t) = tω(1) = t. This contradicts the fact that ω is a nontrivial gauge.

We finish here by proving our last result about the (MAP). We keep the same notation as in Theorem 3.6. Proposition 3.8. Let p ∈ (0, 1) and X be a p-Banach space which admits an FDD with decomposition constant K. Then F (X, • p ) has the K-(BAP). In particular, if X admits a monotne FDD then F (X, • p ) has the (MAP).

Proof. We still denote (M p , d p ) = (X, • p ). Let µ 1 , ..., µ n ∈ F (M p ) and ε > 0. Then there exists N ∈ N and ν 1 , ..., ν n ∈ F (A) (where A = ( N k=1 M p,k , d p )) such that µ k -ν k ≤ ε K . We have seen in the proof of Corollary 3.5 that F (A) has (MAP). Thus, there exists T : F (A) → F (A) a finite rank operator such that T ≤ 1 and T ν k -ν k ≤ ε for every k. Since P N : M p → A is a K-Lipschitz retraction, the linearization PN : F (M p ) → F (A) is projection of norm at most K. This leads us to consider the operator Pn • T : F (M p ) → F (M p ) for which direct computations show that it does the work. Indeed Pn • T is of finite rank, Pn • T ≤ K and for every k:

Pn • T µ k -µ k ≤ Pn • T µ k -Pn • T ν k + Pn • T ν k -ν k + µ k -ν k ≤ Pn • T µ k -ν k + T ν k -ν k + ε ≤ 3ε.

Final comments and open questions

In Proposition 2.6 we stated that if lip 0 (M) is 1-norming then the space F (M) has the Schur property. It is then natural to try to relax the assumption. For instance, we ask the following question Question 4.1. Let M be metric space such that lip 0 (M) is C-norming for some C > 1 ? Does F (M) has the Schur property ? Now remark that we can deduce Proposition 2.16 as a direct consequence of Theorem 2.7 under the additional assumption that F (M) has MAP. However, it seems that most of the examples of proper (in particular compact) metric spaces that we can find in the literature are as follows. Either S 0 (M) separates points uniformly and F (M) has (MAP), or S 0 (M) does not separate points uniformly and F (M) does not have (MAP). So we wonder: Question 4.2. Let M be proper metric space such that S 0 (M) separates points uniformly. Then does F (M) have (MAP) ? In particular, if M is proper and if ω is a nontrivial gauge, does F (M, ω • d) have (MAP) ?

In Section 4, we proved (Proposition 3.6) that Lipschitz-free spaces, over some metric spaces originating from p-Banach spaces, have the Schur property. It is then natural to wonder if we can extend this result to a larger class of metric spaces. Surprisingly, it is really easy to see that we cannot extend this result to every metric space originating from a p-Banach space. Indeed, consider the metric space M originating from L p [0, 1]. Then the map ϕ : t ∈ [0, 1] → ½ [0,t] ∈ M is a nonlinear isometry. Therefore there is a linear and isometric embedding of F ([0, 1]) = L 1 ([0, 1]) into F (M). Consequently, since the Schur property is stable under passing to subspaces and L 1 does not have it, F (M) does not have the Schur property. Furthermore, using the same ideas one can show that F ([0, 1] 2 ) linearly embeds into F (M). Thus, using the fact that F (R 2 ) does not embed into L 1 (see [START_REF] Naor | Planar earthmover is not in L 1[END_REF]), we get that F (M) does not embed into L 1 . A major difference between L p and the p-Banach spaces studied in Section 3 is that L p has trivial dual. In particular, the dual does not separate points of L p . This suggests the following question.

Question 4.3. Consider (X, • ) a p-Banach space whose dual X * separates points. Then does F (X, • p ) have the Schur property ?

  with the obvious extension of the notion of Lipschitz maps). Now the right hand side of the inequality in Lemma 3.1 implies that φ k : (A, • 1 ) → (R, | • |) is n 1-p p -Lipschitz.So we can extend each φ k without increasing the Lipschitz constant and we denote ϕ k those corresponding extensions. Summarizing we have ϕ

Corollary 3 . 4 .

 34 Let p ∈ (0, 1), n ∈ N. Consider (M p , d p ) = (X, • p ) where (X, • ) is a p-Banach space of finite dimension. Then, the space F (M p ) is isometric to S 0 (M p ) * . In particular S 0 (M p ) is 1-norming.Proof. Note that since X p is of finite dimension, it is isomorphic to ℓ n p for some n ∈ N. Thus there is a bi-Lipschitz map between M p and M n p , let us say L:M p → M n p is bi-Lipschitz with C 1 d Mp (x, y) ≤ d M n p (L(x), L(y)) ≤ C 2 d Mp (x, y). Now S 0 (M p ) is C 2 C 1 -norming.Indeed pick x = y ∈ M p and ε > 0. Since S 0 (M n p ) is 1-norming there exists f ∈ S 0 (M n p ) with Lipschitz constant less than 1 + ε such that|f (L(x)) -f (L(y))| = d M n p (L(x), L(y)) ≥ C 1 d Mp (x, y). Now f • L: M p → R is Lipschitz with Lipschitz constant less than C 2 (1 + ε).Moreover as the composition of a bi-Lipschitz map with an element of S 0 (M n p ) we know that f • L ∈ S 0 (M p ). Thus S 0 (M p ) is C 2 C 1 -norming. Since M p is proper, it follows again from Theorem 2.3 that F (M p ) is isometric to S 0 (M p ) * and thus S 0 (M p ) is 1-norming.

Acknowledgments. The author is grateful to G. Lancien and A. Procházka for useful conversations. Finally, the author is grateful to the referee for useful comments and suggestions which permitted to improve the presentation of this paper.


Aris Daniilidis

Colin Petitjean

A partial answer to the Demyanov-Ryabova conjecture 1

Keywords: Polytope, extreme point, sublinear function, subdifferential, exhauster AMS Subject Classification Primary 52B12 ; Secondary 49J52, 90C49

In this work we are interested in the Demyanov-Ryabova conjecture for a finite family of polytopes. The conjecture asserts that after a finite number of iterations (successive dualizations), either a 1-cycle or a 2-cycle eventually comes up. In this work we establish a strong version of this conjecture under the assumption that the initial family contains "enough minimal polytopes" whose extreme points are "well placed".

Introduction

We call polytope any convex compact subset of R N with a finite number of extreme points. Throughout this work we consider a finite family ℜ = {Ω 1 , . . . , Ω ℓ } of polytopes of R N together with an operation which transforms the initial family ℜ to a dual family of polytopes that we denote F(ℜ). (Motivation and origin of this operation will be given at the end of the introduction).

Let us now describe the operation F: let ext(Ω) stand for the set of extreme points of the polytope Ω and let S denote the unit sphere of R N . Then given a family ℜ as before, for any direction d ∈ S and polytope Ω i ∈ ℜ (i ∈ {1, . . . , ℓ}) we consider the set of d-active extreme points of Ω i E(Ω i , d) := {x ∈ ext(Ω i ) : x, d = max Ω i , d }.

We associate to d ∈ S the polytope

Ω(d) := conv   Ω i ∈ℜ E(Ω i , d)   , (1.1)
that is, the polytope obtained as convex hull of the set of all d-active extreme points (when Ω i is taken throughout ℜ). Since the set of extreme points of all polytopes of the family ℜ

E ℜ = Ω i ∈ℜ ext(Ω i ) (1.2)
is finite, the family of polytopes

F(ℜ) := {Ω(d) : d ∈ S} (1.3)
is also finite, hence of the same nature as ℜ. We call F(ℜ) the dual family of ℜ.

Now starting from a given family of polytopes ℜ 0 , we define successively a sequence of families {ℜ n } n by applying repeatedly this duality operation (transformation) F, that is, setting ℜ n+1 := F(ℜ n ), for all n ∈ N. Since the transformation F cannot create new extreme points, the sequence

E ℜn = Ω∈ℜn ext(Ω) (extreme points of polytopes in ℜ n ) n ∈ N
is nested (decreasing) and eventually becomes stable, equal to a finite set E. By a standard combinatorial argument, we now deduce that for some k ≥ 1 and n 0 ≥ 0 we necessarily get ℜ n = ℜ n+k (and E ℜn = E), for all n ≥ n 0 . Therefore, a k-cycle (ℜ n 0 , ℜ n 0 +1 , • • • , ℜ n 0 +k-1 ) is always formed. We are now ready to announce the conjecture of Demyanov and Ryabova:

• Conjecture (Demyanov-Ryabova, [START_REF] Demyanov | Exhausters, coexhausters and converters in nonsmooth analysis[END_REF]). Let ℜ 0 be a finite family of polytopes in R N .

Then for some n 0 ∈ N we shall have ℜ n 0 = ℜ n 0 +2 .

In other words, after some threshold n 0 the sequence

ℜ 0 , ℜ 1 = F(ℜ 0 ), • • • , ℜ n+1 = F(ℜ n ), • • • stabilizes to either a 1-cycle (self-dual family ℜ n = F(ℜ n ) = ℜ n+1 ) or to a 2-cycle (reflexive family ℜ n = F(F(ℜ n )) = ℜ n+2 ) for n ≥ n 0 .
In [START_REF] Demyanov | Exhausters, coexhausters and converters in nonsmooth analysis[END_REF], the authors carried out generic numerical experiments over two hundred families of polytopes, where only 2-cycles eventually arise. Notice however that one can construct particular examples where a 1-cycle is formed. In all known cases, the initial family ℜ 0 ends up, after finite iterations, to a reflexive one.

Besides the recorded numerical evidence, there is still no proof of this conjecture. The only known result in this direction is due to [START_REF] Sang | On the conjecture by Demyanov-Ryabova in converting finite exhausters[END_REF]. In that work, the author establishes the conjecture under the additional assumption that the set E ℜ 0 of extreme points of the initial family ℜ 0 is affinely independent.

Before we state and prove our main result, let us mention that in 1-dimension the conjecture is trivially true.

Proposition 1.1 (The conjecture is true in 1-dim). Let ℜ 0 be a finite family of closed bounded intervals of R. Then ℜ 1 = ℜ 3 .

Proof. Let us denote {I 1 , . . . , I ℓ } the elements of ℜ 0 with I j = [a j , b j ], j ∈ {1, . . . , ℓ}. Since the unit sphere S = S R = {1, -1} consists of only two directions, the construction of the dual family ℜ 1 = F(ℜ 0 ) is very simple. To this end, we set a -:= min i∈{1..ℓ} a i , a + := max i∈{1..ℓ} a i , b -:= min i∈{1..ℓ} b i , b + := max i∈{1..ℓ} b i . This leads to the family

ℜ 1 = {Ω 1 (-1), Ω 1 (1)} = {[a -, a + ], [b -, b + ]}.
The construction of ℜ 2 = F(ℜ 1 ) is even simpler, since we only have two intervals (polytopes) to consider. We actually have

ℜ 2 = {Ω 2 (-1), Ω 2 (1)} = {[a -, b -], [a + , b + ]}.
It now suffices to compute ℜ 3 and obtain directly that ℜ 1 = ℜ 3 . (Notice that if it happens a + = b -then we actually get a 1-cycle:

ℜ 1 = ℜ 2 .)
The extreme simplicity of the problem in dimension 1 is due to the fact that the family that arises after any new iteration has at most 2 elements (corresponding to the directions 1 and -1 of the unit sphere S R ). The problem gets much more complicated though in higher dimensions, where no prior efficient control on the cardinality of the iterated families can be obtained (apart from an absolute combinatorial bound on the number of all possible polytopes that can be obtained by convexifying subsets of the prescribed set of extreme points E). We shall now treat this general case.

Let ℜ 0 be a finite family of polytopes in R N (N ≥ 2). We denote by E := E ℜ 0 the set of extreme points of all polytopes of the family, see (1.2), by R = card(E) its cardinality and we set C := conv(E) its convex hull. Notice that every polytope Ω of the family ℜ 0 (or of any family ℜ n obtained after n-iterations, for every n ∈ N), is contained in C. Let further

r(Ω) := card(Ω ∩ E)
denote the number of extreme points of the polytope Ω ∈ ℜ 0 and set

r min := min Ω∈ℜ 0 r(Ω). (1.4)
We now state the main result of the paper.

Theorem 1.2 (Main result). Let ℜ 0 be a finite family of polytopes in R N and r min ∈ {1, . . . , R} as in (1.4). Then ℜ 1 = ℜ 3 (i.e. a reflexive family occurs after one iteration) provided:

(H1) ∀ x ∈ E, x ∈ conv(E\{x}) (i.e. each x ∈ E is extreme in C.) (H2) ℜ 0 contains all r min -polytopes (that is, all polytopes made up of r min points of E).

Remark 1.3. (i) Assumption (H 1 ) easily yields that the set of extreme points remains stable from the very beginning, that is,

E ℜn = E ℜ 0 = E, for all n ∈ N.
Indeed pick x ∈ E and e x ∈ S which exposes x in C. Let Ω ∈ ℜ 0 be such that x ∈ ext(Ω) (there is clearly at least one such a polytope in ℜ 0 ). Then e x exposes x in Ω, that is x ∈ E(Ω, e x ). It follows readily that x ∈ Ω(e x ) ⊂ E ℜ 1 (see the definition in (1.1)) and by a simple induction,

x ∈ E ℜn , for every n ≥ 1.

(ii) Assumption (H 2 ) will be weakened in the sequel.

Origin of the conjecture. The initial motivation which eventually led to the formulation of the above conjecture stems from the problem of stable representation of positively homogeneous polyhedral functions as finite minima of sublinear ones, or its geometric counterpart, the representation of a closed polyhedral cone as finite union of closed convex polyhedral cones. Let us recall that a function f : R N → R is called positively homogeneous provided f (λx) = λf (x) for every x ∈ R N and λ > 0. It is called sublinear (respectively, superlinear) if it is positively homogeneous and convex (respectively, concave).

Following [START_REF] Pshenichnyi | Convex Analysis and Extremal Problems[END_REF], a sublinear function g : R N → R is called an upper convex approximation of f if g majorizes f on R N , that is, g(x) ≥ f (x), for every x ∈ R N . In the same way, a superlinear function g: R N → R is called a lower concave approximation of f if g minorizes f on R N , that is, g(x) ≤ f (x) for all x ∈ R N . Then we say that a set of sublinear functions E * is an upper exhaustive family for f if the following equality holds for every x ∈ R N :

f (x) = inf g∈E * g(x). (1.5)
Similarly, we say that a set of superlinear functions E * is a lower exhaustive family for f if the following equality holds for every x ∈ R N :

f (x) = sup g∈E * g(x). (1.6)
In [START_REF] Demyanov | Negladkie zadachi teorii optimizatsii i upravleniya[END_REF] the authors established the existence of an upper exhaustive family of upper convex approximations (respectively lower exhaustive family of lower concave approximations) when f is upper semicontinuous on R N (respectively lower semicontinuous). In particular, if f is continuous, the existence of both such families is guaranteed.

It is well known (see [START_REF] Hiriart-Urruty | Fundamentals of convex analysis[END_REF][START_REF] Phelps | Convex functions, monotone operators and differentiability[END_REF] e.g.) that a function g: R N → R is sublinear if and only if g(x) = max h∈∂g(0) x, h . Using this fact we are able to restate (1.5) in the following way:

f (x) = inf g∈E * g(x) = inf g∈E * max h∈∂g(0) x, h = inf Ω∈ℜ max h∈ Ω x, h ,
where ℜ = {∂g(0) : g ∈ E * } is the family of subdifferentials of the sublinear functions g that represent f and Ω = ∂g(0). In a similar way, considering superlinear functions g (lower concave approximations of f ) and denoting by ℜ = {-∂(-g)(0) : g ∈ E * } the family of superdifferentials Ω = -∂(-g)(0), we can restate (1.6) as follows:

f (x) = sup Ω∈ℜ min h∈Ω x, h .
In case of a polyhedral function f the exhaustive families E * and E * can be taken to be finite, with elements being polyhedral functions (g and g respectively). In this case, the corresponding families ℜ and ℜ -called upper (respectively lower) exhausters-are made up of finite polytopes. In [START_REF] Demyanov | Exhausters, coexhausters and converters in nonsmooth analysis[END_REF], the authors presented a procedure -that they called converter -which permits to define from a given lower exhauster ℜ an upper exhauster ℜ = F(ℜ) and vice-versa (this is actually the same procedure and coincides with the described operator F in the beginning of the introduction). A lower (respectively, an upper) exhauster

ℜ (respectively, ℜ) is called stable or reflexive, if ℜ = F (F(ℜ)) (respectively, ℜ = F F(ℜ) ).
An equivalent way to formulate the Demyanov-Ryabova conjecture is to assert that starting with any finite (upper or lower) exhaustive family of polyhedral functions, we eventually end up to a stable one.

Preliminary results

Notation. ℜ 0 is a finite set of polytopes in R N with N ≥ 2. S denotes the unit sphere of R N . ext(Ω) is the set of extreme points of a given polytope Ω ∈ ℜ 0 . E = Ω∈ℜ 0 ext(Ω) is the set of extreme points of all polytopes in ℜ 0 . R := card(E)

C := conv(E)
We assume throughout the paper that E satisfies the assumption (H1) of Theorem 1.2. For the proof of Theorem 1.2, we shall need the two following notions.

Definition 2.1.

• (d-compatible enumeration) An enumeration {x i } R i=1 of E is called d-compatible with respect to a direction d ∈ S, provided x 1 , d ≤ x 2 , d ≤ • • • ≤ x R , d . (2.1)
Notice that a d-compatible enumeration is not necessarily unique: indeed, whenever x i , d = x j , d , for 1 ≤ i < j ≤ R the elements x i and x j can be interchanged in the above enumeration.

• (strict p-location) A direction d ∈ S is said to locate strictly an element x ∈ E at the p-position (where p ∈ {1, . . . , R}), if there exists a d-compatible enumeration {x i } R i=1 of E for which x p = x and . . .

≤ x p-1 , d < x p , d < x p+1 , d ≤ . . .
In case p = 1 (resp. p = R) the left strict inequality x p-1 , d < x p , d (resp. the right strict inequality x p , d < x p+1 , d ) is vacuous. Notice further that since C is a polytope, assumption (H 1 ) yields that for every x ∈ E the normal cone

N C (x) = {d ∈ R N : d, y -x ≤ 0, ∀y ∈ C}
of C at x has nonempty interior (see [START_REF] Hiriart-Urruty | Fundamentals of convex analysis[END_REF][START_REF] Rockafellar | Convex analysis[END_REF] e.g.), and every d ∈ S ∩ int N C (x) strictly locates x in the R-position, under any d-compatible

enumeration {x i } R i=1 of E. • (selection) A map x ∈ E → e x ∈ S is called a selection if ∀x ∈ E, e x ∈ S ∩ int N C (x).
Thus, for every x ∈ E, e x is a direction that strictly exposes x.

We now begin a series of "reordering results". The main goal is the following. Given a d-compatible enumeration of E which locates an element x at some position, say i, we construct a direction d ′ ∈ S and a d ′ -compatible enumeration of E which locates strictly x to a possibly different position p ≥ i. To construct such a d ′ , the general idea is to do small perturbations on d using other well chosen directions. These perturbations need to be quantified and adequately controlled. We start with the following simple lemma. (ii) For t > 0 (respectively t < 0) large enough in absolute value, any (D x (t)/ D x (t) )compatible enumeration (x i ) R i=1 of E strictly locates x at the R-position (resp. at the 1-position). That is, for every y ∈ E, y = x: x, D x (t) > y, D x (t) (resp. x, D x (t) < y, D x (t) ).

(iii) For every y 1 , y 2 ∈ E:

| y 1 -y 2 , D x (t) -D x (0) | ≤ M |t|. (iv) For every y ∈ E, y = x: | x -y, D x (t) -D x (0) | ≥ m|t|.
Proof. The first assertion is obvious. The second assertion is a simple consequence of the fact that e x ∈ int N C (x) exposes x. Now let us prove (iii). We define M = max{ y 1y 2 : y 1 , y 2 ∈ E} > 0. Then, for every

y 1 , y 2 ∈ E, | y 1 -y 2 , D x (t) -D x (0) | = | y 1 -y 2 , te x | ≤ |t| y 1 -y 2 e x ≤ M |t|.
In the same way we prove (iv). Define m = min{| xy, e x | : x, y ∈ E, x = y} > 0. Then for every x, y ∈ E with y = x,

| x -y, D x (t) -D x (0) | = |t| | x -y, e x | ≥ m|t|.
Remark 2.3. Note that, whenever the selection x ∈ E → e x ∈ S ∩ int N C (x) is fixed, the constants m and M in the previous lemma hold for every function D x (and do not depend neither on d, nor on x).

The next lemma will play a key role in the sequel.

Lemma 2.4 (Strict location in the very next position). Let {x j } R j=1 be a d-compatible enumeration of E such that for some 1 ≤ i ≤ R -1 we have:

. . . ≤ x i-1 , d < x i , d ≤ x i+1 , d ≤ . . .
Then there exist a direction d ′ ∈ S and a d ′ -compatible enumeration {y j } R j=1 satisfying

{x 1 , • • • , x i-1 } ⊂ {y 1 • • • , y i }
and locating strictly x i at the i + 1-position, that is,

y i+1 = x i . . . ≤ y i , d ′ < y i+1 , d ′ < y i+2 , d ′ ≤ . . .
Proof. Throughout the proof, we fix x ∈ E → e x ∈ S ∩ int N C (x) a selection and m, M > 0 the universal constants given in Lemma 2.2 (c.f. Remark 2.3).

Case 1: x i is not strictly located in the i-position, that is the d-compatible enumeration

{x j } R j=1 verifies . . . ≤ x i-1 , d < x i , d = x i+1 , d ≤ . . .
An additional difficulty here is that they may exist more than one y ∈ E such that x i , d = y, d (that is x i+1 may not be the unique point with this property). So let k ∈ {i -1, . . . , R} be the maximum index such that x i , d = x k , d . Our strategy would be to do a small perturbation on d with a good control in order to put x i at the i-position strictly. Of course this creates a new direction d ′ together with a new ordering of elements in E through d ′ . Then, we consider an element y which is right after x i in the d ′ -ordering. Again, we do a small perturbation of d ′ with a good control in order to reverse the order of x i and y. The key point is the uniform control of the employed perturbations ensuring that the element x i reaches the i + 1-position and not a further position.

Let us write a = x ix i-1 , d > 0, c = M/m and let ε > 0 such that a -2cε > a/2 > 0. Let us summarize our notations with the following picture

✲ •, d x 1 • • • x i-1 x i x i+1 • • • x R ✲ ✛ a
Step 1: We locate x i strictly in the i-position but in a controlled way. Consider the map D x i (t) = d + te x i defined in Lemma 2.2, and then define the function

Φ : t ∈ R → min j∈{i+1,...,R} x j -x i , D x i (t) .
The map Φ is continuous, satisfies Φ(0) = 0 and lim t→-∞ Φ(t) = +∞. Thus, by the intermediate value theorem, there exists t 0 < 0 such that Φ(t 0 ) = ε. That is min j∈{i+1,...,R}

x j , D x i (t 0 ) = x i , D x i (t 0 ) + ε.
Taking ε > 0 small enough we ensure that if y ∈ (x i ) R j=i+1 is such that yx i , D x i (t 0 ) = ε, then y ∈ {x i+1 , . . . , x k }. Pick such a y ∈ (x i ) k j=i+1 . Thanks to the assertion (iv) of Lemma 2.2, we have

ε = | y -x i , D x i (t 0 ) -D x i (0) | ≥ m|t 0 |.
Thus |t 0 | ≤ ε/m. Next, thanks to the assertion (iii) of Lemma 2.2, for every j in {1, . . . , i -1} we have:

| x i -x j , D x i (t 0 ) -D x i (0) | ≤ M |t 0 | ≤ cε.
This implies that

x i -x j , D x i (t 0 ) ≥ x i -x j , D x i (0) -cε ≥ a -cε. (2.2)
Therefore we obtain a (D

x i (t 0 )/ D x i (t 0 ) )-compatible enumeration (x ′ i ) R i=1 satisfying {x 1 , . . . , x i-1 } = {x ′ 1 , . . . , x ′ i-1 }, x i = x ′ i and y = x ′ i+1 ∈ {x i+1 , . . . , x k }.
We resume the situation in the following picture:

✲ •, D x i (t 0 ) x ′ 1 • • • x ′ i-1 x ′ i x ′ i+1 • • • x ′ R ✲ ✛ ≥ a -cε ✲ ✛ ε
Step 2: We define a new direction d together with a d-enumeration which locates x i at the (i + 1)-position and such that there is only one element y in E, y = x i , verifying

x i , d = y, d . Consider D x ′ i+1 (t) = D x i (t 0 ) + te x ′ i+1 .
Reasoning as before, by the intermediate value theorem, there exists t 1 < 0 such that x ′ i+1x ′ i , D x ′ i+1 (t 1 ) = 0. Thanks to the assertion (iv) of Lemma 2.2, we have

ε = | x ′ i+1 -x ′ i , D x ′ i+1 (t 1 ) -D x ′ i+1 (0) | ≥ m|t 1 |. Thus |t 1 | ≤ ε/m.
Next, thanks to the assertion (iii) of Lemma 2.2, evoking 2.2 under the new enumeration {x ′ i } R i=1 , for every j ∈ {1, . . . , i -1} we deduce:

x ′ i -x ′ j , D x ′ i+1 (t 1 ) ≥ x ′ i -x ′ j , D x ′ i+1 (0) -M |t 1 | ≥ (a -cε) -cε = a -2cε.
Note that we also have

x ′ j -x ′ i+1 , D x ′ i+1 (t 1 ) ≥ m|t 1 | for j ≥ i + 2. Therefore, denot- ing d := D x ′ i+1 (t 1 )/ D x ′ i+1 (t 1 ) , we may fix (x ′′ i ) R i=1 a d-compatible enumeration satisfying {x ′′ 1 , . . . , x ′′ i-1 } = {x 1 , . . . , x i-1 }, x ′′ i = x ′ i+1 and x ′′ i+1 = x ′ i = x i .
This leads us to the following configuration:

✲ •, D x i+1 (t 1 ) x ′′ 1 • • • x ′′ i-1 x ′′ i x ′′ i+1 x ′′ i+2 • • • x ′′ R ✲ ✛ ≥ a -2cε ✲ ✛ ≥ m|t 1 |
Step 3: Conclusion. To complete the proof. It suffices to evoke a continuity argument and take t 2 ∈ (-∞, t 1 ) such that:

x ′′ i , D x ′ i+1 (t 2 ) > x ′′ j , D x ′ i+1 (t 2 ) , ∀ j ∈ {1, . . . , i -1} x ′′ i+1 , D x ′ i+1 (t 2 ) > x ′′ i , D x ′ i+1 (t 2 ) x ′′ ℓ , D x ′ i+1 (t 2 ) > x ′′ i+1 , D x ′ i+1 (t 2 ) , ∀ ℓ ∈ {i + 2, . . . , R}. Setting d ′ = D x ′ i+1 (t 2 )/ D x ′ i+1 (t 2 )
, we deduce the existence of a d ′ -compatible enumeration {y j } R j=1 satisfying the desired conditions. That is

{y 1 • • • , y i-1 } = {x ′′ 1 , • • • , x ′′ i-1 } = {x 1 , • • • , x i-1 }, y i = x ′′ i , y i+1 = x ′′ i+1 = x i and . . . ≤ y i , d ′ < y i+1 , d ′ < y i+2 , d ′ ≤ . . .
This finishes the first part of the proof.

Case 2: x i is strictly located in the i-position, that is x i , d < x i+1 , d . We prove that this case reduces to the first case. Indeed, consider D x i : t → d + te x i the map given by Lemma 2.2. Applying again the intermediate value theorem we deduce the existence of t 0 > 0 such that

x i , D x i (t 0 ) = min j∈{i+1,...,R} x j , D x i (t 0 ) .
Thus, replacing d by d := D x i (t 0 )/ D x i (t 0 ) we obtain a d-compatible enumeration (y j ) R i=1 of E verifying {x 1 , . . . , x i-1 } ⊂ {y 1 , . . . , y i-1 }, y i = x i and y i , d ′ = y i+1 , d ′ . Therefore we rejoined the first case.

Remark 2.5. It might seem strange, at a first sight, to back to the first case, since the first step of the latter was precisely to apply a perturbation that strictly locates x i in the i-position. However, as we pointed out in the proof, this is done in a precise quantified way.

The following corollary is an easy consequence of the previous lemma and will be recalled in several occasion in the proof of Theorem 1.2.

Corollary 2.6 (Reordering lemma). Let {x i } R i=1 be a d-compatible enumeration of E and assume that x i , d < x p , d for 1 ≤ i < p ≤ R. Then there exist a direction d ′ ∈ S and a d ′compatible enumeration

{y j } R j=1 satisfying {x 1 , • • • , x i-1 } ⊆ {y 1 , • • • , y p-1 }
and strictly locating x i at the p-position, that is,

y p = x i . . . ≤ y p-1 , d ′ < y p , d ′ < y p+1 , d ′ ≤ . . .
Proof. First note that if x i-1 , d < x i , d , then the result follows from Lemma 2.4 applied successively pi times. So let us assume that x i-1 , d = x i , d . Fix e x i ∈ intN C (x i ) and consider the map D x i : t ∈ R → d + te x i given by Lemma 2.2. Recall that D x i is continuous with D x i (0) = d. Since x i , d < x p , d , there exists t 0 > 0 such that x i , D x i (t 0 ) < x p , D x i (t 0 ) and x i is strictly located at some position, say k, in every (D x i (t 0 )/ D x i (t 0 ) )-compatible enumeration. Thus we set d := D x i (t 0 )/ D x i (t 0 ) and we fix (

x ′ i ) R i=1 a d-compatible enumeration. Of course we have {x 1 , • • • , x i-1 } ⊆ {x ′ 1 , • • • , x ′ k-1 }
R n = F(R n-1 ) = {Ω n (d) : d ∈ S}. (3.1)
We recall that a subset F of a polytope Ω is called a face of Ω if there exists a direction d ∈ S such that

F = {x ∈ Ω : x, d = min z∈Ω z, d }. (3.2)
In this case we denote the face by F (Ω, d). Notice that for any d ∈ S it holds:

F (C, d) = [E(C, -d)].
We are ready to proceed to the proof of Theorem 1.2.

Proof of Theorem 1.2. In view of Proposition 1.1 we may assume N ≥ 2. Let us first treat the case r min = 1, that is, the case where the initial family ℜ 0 contains all singletons. In this case, pick any x ∈ E and d ∈ S. Since Ω x = [x] ∈ ℜ 0 , we deduce that E(Ω x , d) = {x} and consequently, x ∈ Ω 1 (d). It follows that Ω

1 (d) = [E] = C for all d ∈ S, that is, ℜ 1 = {C}.
Consequently, the family ℜ 2 consists of all faces of C, that is,

ℜ 2 = {[E(C, d)] : d ∈ S} ≡ {F (C, d) : d ∈ S}.
In particular, for x ∈ E and d ∈ int N C (x) (direction that exposes x in C) we get F (C, -d) = [x], therefore ℜ 2 contains all singletons and ℜ 3 = {C} = ℜ 1 .

Let us now treat the case r min = R. In this case ℜ 0 = {C} and we deduce, as before, that ℜ 1 is the family of all faces of C and ℜ 2 = {C} = ℜ 0 . It remains to treat the case r min / ∈ {1, R} which is what we assume in the sequel. In this case, we show that ℜ 1 = ℜ 3 which in view of (3.1) yields

F(ℜ 0 ) = F(ℜ 2 )), i.e. Ω 1 (d) = Ω 3 (d) for every d ∈ S. (3.3)
To establish (3.3) we shall proceed in three steps (Subsections 3.1-3.3), characterizing respectively, the polytopes belonging to the families ℜ 1 , ℜ 2 and respectively ℜ 3 .

3.1 Characterization of polytopes in ℜ 1 .

In this step, by means of geometric conditions on C we characterize membership of a given polytope to the family ℜ 1 . We start with the biggest possible polytope, namely C.

Proposition 3.1. Assume ℜ 0 satisfies (H 1 ), (H 2 ). Then the following are equivalent:

(i) C = Ω 1 (d 0 ), for some d 0 ∈ S (that is, C ∈ ℜ 1 = F(ℜ 0 )) ; (ii) card (F (C, d 0 ) ∩ E) ≥ r min (
that is, C has a face containing at least r min points).

Proof. [(ii)=⇒(i)] Let us first assume that for d 0 ∈ S assertion (ii) holds and let us prove that

Ω 1 (d 0 ) = [ Ω∈ℜ 0 E(Ω, d 0 )] = C. (3.4)
It suffices to prove that for each x ∈ E there exists a polytope Ω ∈ ℜ 0 such that x ∈ E(Ω, d 0 ). Since F (C, d 0 ) contains at least r min -1 extreme points different than x, by assumption (H 2 ) the family ℜ 0 contains the polytope Ω obtained by convexification of x and the aforementioned r min -1 points. Recalling (3.2) we deduce

E(Ω, d 0 ) = {x}, if x / ∈ F (C, d 0 ) Ω ∩ E, if x ∈ F (C, d 0 ).
In all cases x ∈ E(Ω, d 0 ) ⊂ Ω 1 (d 0 ), which shows that (3.4) holds true.

[(i)=⇒(ii)] Let us now assume that C = Ω 1 (d 0 ), for some d 0 ∈ S, and let

{x i } R i=1 be a d 0 -compatible enumeration. Let k = max{i : x i , d 0 = x 1 , d 0 } so that F (C, d 0 ) = [x 1 , . . . , x k ].
Assume towards a contradiction, that k < r min , and fix i 0 ∈ {1, . . . , k}. Then (in view of the definition of r min , see (1.4)) any polytope Ω ∈ ℜ 0 that contains x i 0 should necessarily contain some element x j with j > k. In particular, x i 0 , d 0 < x j , d 0 , hence x i 0 / ∈ E(Ω, d 0 ). Thus x i 0 /

∈ Ω 1 (d 0 ), contradicting (i).

Let us now characterize membership of smaller polytopes to ℜ 1 .

Proof. This fact is obvious since ℜ 0 contains all possible r min -polytopes. In particular, there exists a polytope Ω entirely contained in [x 1 , . . . , x k ], and consequently for every d ∈ S it holds E(Ω, d) ∩ [x 1 , . . . , x k ] = ∅.

The proof is complete.

To resume the above results, we have established that a polytope Ω belongs to the family ℜ 1 if and only if there is a d k -compatible enumeration

{x ′ i } R i=1 of E such that Ω = [E {x ′ 1 , . . . , x ′ k }] (0 ≤ k < r min )
with the obvious abuse of notation: k = 0 =⇒ {x ′ 1 , . . . , x ′ k } = ∅.

3.2 Characterization of polytopes in ℜ 2 .

In this step, we shall describe the elements of the family

Lemma 2 . 2 (

 22 Uniform control). Let d ∈ S and fix x ∈ E → e x ∈ S ∩ int N C (x) a selection. Then, there exist constants M > 0 and m > 0 such that, for every x ∈ E, the map D x : R → R N defined for every t ∈ R by D x (t) = d + te x satisfies the following properties: (i) D x is continuous and D x (0) = d.

  and x i is strictly located at the k-position in this d-compatible enumeration. Now the result follows from Lemma 2.4 applied pk times. 3 Proof of the main result (Theorem 1.2) Extra notation. We keep the notation introduced at the beginning of Section 2. For the needs of the proof, we introduce some extra notation. Given E 1 ⊂ E we shall often use the abbreviate notation [E 1 ] = conv(E 1 ). Under this notation we trivially have C = [E]. Starting from a finite family of polytopes R 0 , we recall that R n = F n (R 0 ) (n ≥ 1) where F n means applying the operator F defined in (1.3) n times. For d ∈ S and n ≥ 1 we denote Ω n (d) = [ P ∈ℜ n-1 E(P, d) ], where E(P, d) = {x ∈ ext(P ) : x, d = max P, d }. Under this notation,

ℜ 2 = 1 E. 7 ) 3 . 4 . 1 E

 217341 F(ℜ 1 ) = {Ω 2 (d) : d ∈ S} where as usual, Ω 2 (d) = [ Ω∈ℜ (Ω, d)].Let us proceed to a complete description of the above elements. To this end, let us fix a direction d 0 ∈ S. By the previous step (Subsection 3.1), there exists a d 0 -compatible enumeration{x ′ i } R i=1 of E and k ∈ {0, . . . , r min -1} such that Ω 1 (d 0 ) = [ Ω∈ℜ 0 E(Ω, d 0 )] = [E {x ′ 1 , . . . , x ′ k }] ∈ ℜ 1 . (3Proposition Let {x ′ i } R i=1 denote the above d 0 -compatible enumeration of E for which (3.7) holds. Then Ω 2 (-d 0 ) := [ Ω∈ℜ (Ω, -d 0 )] = [x ′ 1 , . . . , x ′ ℓ ] ∈ ℜ 2 ,where ℓ = max{i :x ′ i , d 0 = x ′ r min , d 0 } (∈ {r min , . . . , R}) . (3.8)Proof. Let us first assume k ≥ 1. According to Proposition 3.2, we havex ′ k , d 0 < x ′ k+1 , d 0 = x ′ r min , d 0 = x ′ ℓ , d 0 = a.Since Ω 1 (d 0 ) ∈ ℜ 1 the above yieldsE(Ω 1 (d 0 ), -d 0 ) = {x ′ k+1 , . . . , x ′ ℓ }, therefore {x ′ k+1 , . . . , x ′ ℓ } ⊂ Ω 2 (-d 0 ). Let further m ∈ {1, . . . , k} be such thatF (C, d 0 ) = [x ′ 1 , . . . , x ′ m ].

Research partially supported by the grants BASAL PFB-03 and ECOS/CONICYT-ECOS/Sud C14E06. The first author has also been partially supported by the grants FONDECYT 1171854 (Chile) and MTM2014-59179-C2-1-P (MINECO of Spain and ERDF of EU)

Acknowledgments. The authors thank Vera Roshchina (University of Melbourne) for introducing to them the conjecture. They also thank Abderrahim Hantoute (CMM, University of Chile) and Bernard Baillon (University Paris 1) for useful discussions. Major part of this work has been accomplished during a research visit of the second author to the Department of Mathematical Engineering of the University of Chile (October 2016) and of the first author to the Laboratory of Mathematics of the University of Franche-Comté in Besançon (December 2016). The authors thank their hosts for hospitality.

Proposition 3.2. Assume ℜ 0 satisfies (H 1 ), (H 2 ). Let x 1 , x 2 , . . . , x k be distinct points in E with 1 ≤ k < r min . The following are equivalent:

and {x 1 , . . . , x k } = {x ′ 1 , . . . , x ′ k }.

(3.6)

Proof. [(ii)=⇒(i)] The proof is very similar to the previous one. Let us first assume that (ii) holds for any 1 ≤ k < r min and distinct points x 1 , x 2 , . . . , x k ∈ E. We shall prove

Pick any i ∈ {1, . . . , k}. Then by (3.6) there exists i 0 ∈ {1, . . . , k} with

r min , d k := α and by assumption, there exist at least r min -1 extreme points with values less or equal to α, forming, together with x an r min -polytope Ω ∈ ℜ 0 for which x ∈ E(Ω, d k ). This shows that

r min , d k and let i 1 ∈ {1, . . . , r min } (respectively, i 2 ∈ {r min , . . . , R}) be the minimum (respectively, maximum) integer i such that

Let us complete this part with the following result.

Proposition 3.3. Assume ℜ 0 satisfies (H 1 ), (H 2 ). Then ℜ 1 does not contain any polytope of the form [E {x 1 , . . . , x k }] where x 1 , . . . , x k ∈ E are distinct and k ≥ r min .

It follows easily that

. Finally, let i ∈ {m + 1, . . . , k} and let us show that x ′ i ∈ Ω 2 (-d 0 ). To this end, we need to exhibit a direction d ′ ∈ S such that the polytope Ω 1 (d ′ ) ∈ ℜ 1 contains x ′ i but does not contain any x ′ j for 1 ≤ j < i. (In such a case we would get

and we are done.) Indeed, let d ′ be given by Corollary 2.6 for p = r min . Then there exists a d ′ -compatible enumeration {y i } R i=1 of E locating strictly x ′ i in the p = r min position (i.e. y r min = x ′ i ) and

and consequently

. Indeed, since ℓ ≥ r min , it follows from Proposition 3.3 that any polytope of ℜ 1 should contain at least one of the elements {x ′ i :

Let us now assume k = 0, that is, Ω 1 (d 0 ) = C. Then according to Proposition 3.1 the face F (C, d 0 ) contains at least r min points of E. In view of (3.8) we deduce that

Using the same argument as before, we get that x ′ j / ∈ Ω 2 (-d 0 ) whenever j ≥ ℓ + 1. Indeed, according to Proposition 3.3, since l ≥ r min any polytope of ℜ 1 should contain at least one of the elements {x ′ i : 1 ≤ i ≤ ℓ}. Thus for any polytope Ω 1 in R 1 containing x j we have x j ∈ E(Ω 1 , -d 0 ). The proof is complete.

Since Proposition 3.4 can be applied to all directions d ∈ S we eventually recover a full description of polytopes in ℜ 2 .

Construction of ℜ 3 and conclusion.

In this part we prove the following assertion: For every d ∈ S, we have Ω 1 (d) = Ω 3 (d). This last statement trivially implies that ℜ 1 = ℜ 3 and finishes the proof of the theorem. Let us proceed to the proof of the assertion. Fix any direction d 0 ∈ S. According to Subsection 3.1, we can fix a d 0 -compatible enumeration (

where k ∈ {0, . . . , r min } (under the convention that {x

, where ℓ ≥ r min being defined in (3.8). Thus, we are in the following configuration:

The above readily yields that

) for all j ∈ {ℓ + 1, . . . , m -1}. Notice that x ′ j is located in the (Rj)-position in the inverse (-d)-compatible enumeration. Applying Corollary 2.6 we obtain a direction (-d ′ ) that pushed forward x j to the (Rr min )-position, locating it there strictly. So we obtain a d ′ -compatible enumeration {y j } R j=1 such that

Writing the above assertion in reverse order yields

and consequently, y r min = x ′ j ∈ E(Ω 1 (d ′ ), d 0 ) ⊂ Ω 3 (d 0 ). It remains to prove that x ′ j ∈ Ω 3 (d 0 ) whenever j ∈ {1, . . . , k}. Indeed, if this were not the case, then there would exist a polytope Ω ∈ ℜ 2 such that x ′ j ∈ E(Ω, d 0 ) and consequently the polytope Ω cannot contain any other element x ∈ E with x, d 0 > x ′ j , d 0 . In particular {x ′ k+1 , . . . , x ′ R }∩Ω = ∅. Thus such a polytope could contain at most k points of E with k < r min , which is impossible according to Proposition 3.4 (every polytope of ℜ 2 contains at least r min points of E). It follows that

which proves the assertion and the theorem.

Weakening assumption (H2)

A careful inspection of the previous proof reveals that some r min -polytopes do not intervene in the construction of the family ℜ 1 = F(ℜ 0 ) and consequently assumption (H 2 ) can be relaxed as follows (we leave the details to the reader):

(H ′ 2 ) The family ℜ 0 contains all r min -polytopes of the form [x 1 , . . . , x r min ] for which there exists a direction d ∈ S and a d-compatible enumeration {x ′ i } R i=1 such that {x 1 , . . . , x r min } = {x ′ 1 , . . . , x ′ r min } and x ′ r min , d < x ′ r min +1 , d .