Pierre Calka

J E Yukich

Variance asymptotics for random polytopes

Keywords: classifications. Primary 60F05, Secondary 60D05 Random convex hulls, parabolic growth and hull processes

come

Introduction

Let K ⊂ R d be a compact convex body with non-empty interior and having a C 3 boundary of positive Gaussian curvature κ. Letting P λ be a Poisson point process in R d of intensity λ we denote by K λ the convex hull of P λ ∩ K. Let f k (K λ ), k ∈ {0, 1, ..., d -1}, be the number of k faces of K λ .

Rényi and Sulanke [START_REF] Rényi | Über die konvexe Hülle von n zufállig gewählten Punkten[END_REF] were the first to consider the average behavior of f 0 (K λ ) in the planar case. Generalizing their formula to higher dimensions, Bárány [START_REF] Bárány | Random polytopes in smooth convex bodies Mathematika[END_REF] showed there is a constant D 0,d such that

lim λ→∞ λ -(d-1)/(d+1) E f 0 (K λ ) = D 0,d ∂K κ(z) 1/(d+1) dz.
The integral ∂K κ(z) 1/(d+1) dz is known as the affine surface area of ∂K. Assuming only that K has a boundary ∂K of differentiability class C 2 , Reitzner [START_REF] Reitzner | The combinatorial structure of random polytopes[END_REF] extended this result to f k (K λ ), k ∈ 1 {0, 1, ..., d -1}, showing for all d ≥ 2 that there are constants D k,d such that lim λ→∞ λ -(d-1)/(d+1) E f k (K λ ) = D k,d ∂K κ(z) 1/(d+1) dz.

(1.1)

Reitzner [START_REF] Reitzner | Central limit theorems for random polytopes[END_REF] also showed that (f k (K λ ) -E f k (K λ ))/ Varf k (K λ ) converges in distribution to a mean zero normal random variable as λ → ∞, though there have been relatively few results

concerning the asymptotic variance of f k (K λ ). Theorem 4 of Reitzner [START_REF] Reitzner | Central limit theorems for random polytopes[END_REF] gives upper and lower bounds of the same magnitude for Varf k (K λ ), k ∈ {0, 1, ..., d -1}, which extends work of Buchta [START_REF] Buchta | An identity relating moments of functionals of convex hulls[END_REF], who obtains lower bounds for Varf 0 (K λ ) of order λ (d-1)/(d+1) . In the special case that K is a ball, closed form variance asymptotics for Varf k (K λ ), k ∈ {0, 1, ..., d -1} are given in [START_REF] Schreiber | Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points[END_REF][START_REF] Calka | Brownian limits, local limits, and variance asymptotics for convex hulls in the ball[END_REF].

Let K ′ n be the convex hull of n i.i.d. random variables uniformly distributed on K. Our main two results resolve the open question of determining variance asymptotics for Varf k (K λ ) and Varf k (K ′ n ), K smooth and convex, as put forth on page 1431 of [START_REF] Weil | Stochastic geometry[END_REF].

Theorem 1.1 For all k ∈ {0, 1, ..., d -1}, there exist positive constants F k,d such that

lim λ→∞ λ -(d-1)/(d+1) Varf k (K λ ) = F k,d ∂K κ(z) 1/(d+1) dz. (1.2)
Let vol be the Lebesgue measure. De-Poissonization methods, based on coupling, yield the following binomial counterpart of (1.2). When k = 0, it resolves Conjecture 1 of Buchta [START_REF] Buchta | An identity relating moments of functionals of convex hulls[END_REF]. Remarks.

(i) Related work. Bárány and Reitzner (page 3 of [START_REF] Bárány | The variance of random polytopes[END_REF]) conjecture for general convex bodies that Varf k (K λ ) should -up to constants -behave like the variance of the volume of the wet part of the floating body, which, in the case of smooth convex sets, is proportional to affine surface area.

Theorem 1.1 resolves a sharpened version of this conjecture in the case that ∂K is smooth.

(ii) The constants F k,d . The proofs of Theorems 1.1 and 1.2 show that F k,d is defined in terms of parabolic growth processes on the upper half-space R d-1 × R + . As noted on page 137 of Buchta [START_REF] Buchta | An identity relating moments of functionals of convex hulls[END_REF], F k,d may also be identified in terms of a constant involving complicated double integrals given in Groeneboom [START_REF] Groeneboom | Limit theorems for convex hulls[END_REF].

(iii) Volume asymptotics. Under a C 3 and C 2 assumption on ∂K, respectively, Bárány [START_REF] Bárány | Random polytopes in smooth convex bodies Mathematika[END_REF] and

Reitzner [START_REF] Reitzner | Stochastic approximation of smooth convex bodies[END_REF] show

lim λ→∞ λ 2/(d+1) E vol(K \ K λ ) = c d (vol(K)) 2/(d+1)
∂K κ(z) 1/(d+1) dz.

(1.4) Böröczky et al. [START_REF] Böröczky | The mean width of random polytopes circumscribed around a convex body[END_REF] extend this limit and (1.1) to convex hulls of i.i.d. points having a non-uniform density on K. Theorem 3 of Reitzner's breakthrough paper [START_REF] Reitzner | Central limit theorems for random polytopes[END_REF] gives upper and lower bounds of the same magnitude for Var vol(K λ ), though it falls short of giving a limiting variance. Notice that Theorems 1.1 and 1.2 fill in this gap as follows. Buchta notes (see Corollary 1 and (3.6) of [START_REF] Buchta | An identity relating moments of functionals of convex hulls[END_REF]) under sufficient smoothness of ∂K, that variance asymptotics for n 2 Varf 0 (K ′ n ) coincide with variance asymptotics for Var vol(K ′ n ), that is

Var vol(K ′ n ) = Var(f 0 (K ′ n+2 )) + d n+2 (n + 1)(n + 2)
,

where lim n→∞ 3 -d d + 1 ∂K κ(z) 1/(d+1) dz • n (d-1)/(d+1) -1 d n = 1.
Consequently, putting G d := F 0,d + (3d)/(d + 1) and putting k = 0 in (1.3), we get

lim n→∞ n (d+3)/(d+1) Var vol(K ′ n ) = G d (vol(K)) (d+3)/(d+1) ∂K κ(z) 1/(d+1) dz. (1.5)
By (1.5) and Proposition 3.2 of [START_REF] Vu | Central limit theorems for random polytopes in a smooth convex set[END_REF], which states that Var vol(K ′ n ) and Var vol(K n ) coincide up to first order, we deduce

lim λ→∞ λ (d+3)/(d+1) Var vol(K λ ) = G d ∂K κ(z) 1/(d+1) dz.
(

The paper is organized as follows. Section 2 introduces the main tool for the proof of Theorem 1.1, namely the paraboloid growth process used in [START_REF] Schreiber | Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points[END_REF] and [START_REF] Calka | Brownian limits, local limits, and variance asymptotics for convex hulls in the ball[END_REF]. We state a general result, Theorem Finally, in Section 6, we prove the de-Poissonized limit (1.3).

Paraboloid growth processes and a general result

Given a finite point set X ⊂ R d , let co(X ) be its convex hull.

Definition 2.1 Given k ∈ {0, 1, ..., d -1} and x a vertex of co(X ), define the k-face functional ξ k (x, X ) to be the product of (k + 1) -1 and the number of k faces of co(X ) which contain x.

Otherwise we put ξ k (x, X ) = 0. The empirical k-face measure is

µ ξ k λ := x∈P λ ∩K ξ k (x, P λ ∩ K)δ x , (2.1)
where δ x is the unit point mass at x.

Thus the number of k-faces in co(X ) is x∈X ξ k (x, X ). We shall give a general result describing the limit behavior of µ ξ k λ in terms of parabolic growth processes on R d .

Paraboloid growth processes. Denote points in R d-1 × R by w := (v, h) or w ′ := (v ′ , h ′ ), depending on context. Let Π ↑ be the epigraph of the parabola v → |v| 2 /2, that is Π ↑ := {(v, h) ∈ R d-1 × R + , h ≥ |v| 2 /2}. Letting X ⊂ R d be locally finite, define the parabolic growth model Ψ(X ) := w∈X (w ⊕ Π ↑ ),
where ⊕ denotes Minkowski addition. A point w 0 ∈ X is extreme with respect to Ψ(X ) if the epigraph w 0 ⊕ Π ↑ is not a subset of the union of the epigraphs {w ⊕ Π ↑ , w ∈ X \ w 0 }, that is

(w 0 ⊕ Π ↑ ) w∈X \w0 (w ⊕ Π ↑
). The paraboloid hull model Φ(X ) is defined as in Definition 3.4 of [START_REF] Calka | Brownian limits, local limits, and variance asymptotics for convex hulls in the ball[END_REF]:

Φ(X ) := w∈R d-1 ×R (w⊕Π ↓ )∩X =∅ (w ⊕ Π ↓ ), where Π ↓ := {(v, h) ∈ R d-1 × R, h ≤ -|v| 2 /2}.
It may be viewed as the dual of the paraboloid growth model Ψ(X ). Let P be a rate one homogeneous Poisson point process on R d-1 × R + and let Ψ := Ψ(P) and Φ := Φ(P) be the corresponding paraboloid growth and hull processes. As in [START_REF] Calka | Brownian limits, local limits, and variance asymptotics for convex hulls in the ball[END_REF], the set Vertices(Φ) coincides with the extreme points of Ψ. Definition 2.2 (cf. section 6 of [START_REF] Calka | Brownian limits, local limits, and variance asymptotics for convex hulls in the ball[END_REF]). Define the scaling limit k-face functional ξ (∞) k (x, P), for x ∈ P, and k ∈ {0, 1, ..., d-1}, to be the product of (k+1) -1 and the number of k-dimensional paraboloid faces of the hull process Φ which contain x, if x belongs to Vertices(Φ), and zero otherwise.

One of the main features of our approach is that ξ (∞) k is indeed a scaling limit of appropriately re-scaled k-face functionals, as seen in Lemma 4.6 of Section 4 and also in Lemma 7.2 of [START_REF] Calka | Brownian limits, local limits, and variance asymptotics for convex hulls in the ball[END_REF].

Define the following second order correlation functions for ξ (∞) (x, P) := ξ (∞) k (x, P) (cf. (7.2), (7.3) of [START_REF] Calka | Brownian limits, local limits, and variance asymptotics for convex hulls in the ball[END_REF]).

Definition 2.3 For all w 1 , w 2 ∈ R d , put ζ ξ (∞) (w 1 ) := ζ ξ (∞) (w 1 , P) := E ξ (∞) (w 1 , P) 2 (2.2)
and

ζ ξ (∞) (w 1 , w 2 ) := ζ ξ (∞) (w 1 , w 2 , P) := (2.3) E ξ (∞) (w 1 , P ∪ {w 2 })ξ (∞) (w 2 , P ∪ {w 1 }) -E ξ (∞) (w 1 , P)E ξ (∞) (w 2 , P).
Note that

σ 2 (ξ (∞) ) := ∞ 0 ς ξ (∞) ((0, h))dh + ∞ 0 R d-1 ∞ 0 ς ξ (∞) ((0, h), (v ′ , h ′ ))dh ′ dv ′ dh (2.4)
is finite and positive by Theorems 7.1 and 7.3 in [START_REF] Calka | Brownian limits, local limits, and variance asymptotics for convex hulls in the ball[END_REF].

Theorem 1.1 is a special case of the following more general result giving the asymptotic behavior of the empirical k-face measures in terms of parabolic growth processes. Let C(K) be the class of continuous functions on K and let g, µ ξ λ denote the integral of g with respect to µ ξ λ .

Theorem 2.1 For all g ∈ C(K) and k ∈ {0, 1, ..., d -1}, we have

lim λ→∞ λ -(d-1)/(d+1) E [ g, µ ξ k λ ] = ∞ 0 E ξ (∞) k ((0, h), P)dh ∂K g(z)κ(z) 1/(d+1) dz (2.5)
and ).

lim λ→∞ λ -(d-1)/(d+1) Var[ g, µ ξ k λ ] = σ 2 (ξ (∞) k ) ∂K g(z) 2 κ(z)
(iii) Extensions. As in [START_REF] Reitzner | Central limit theorems for random polytopes[END_REF] and [START_REF] Böröczky | The mean width of random polytopes circumscribed around a convex body[END_REF], we expect that the C 3 boundary condition could be relaxed to a C 2 condition, and we comment on this in Section 5.3. Following the methods of Section 6, we obtain the counterpart of Theorem 2.1 for binomial input.

Affine and scaling transformations

For each z ∈ ∂K, we first consider an affine transformation A z of K, one under which the scores ξ k are invariant, but under which the principal curvatures of A z (K) at z coincide, that is to say A z (K) is 'umbilic' at z. This property allows us to readily approximate the functionals ξ k on Poisson points in A z (K) by the corresponding functionals on Poisson points in the 'osculating ball' at z, defined below. The key idea of replacing the mother body K with an osculating ball has been used by Rényi and Sulanke [START_REF] Rényi | Über die konvexe Hülle von n zufállig gewählten Punkten II[END_REF], Bárány [START_REF] Bárány | Random polytopes in smooth convex bodies Mathematika[END_REF], and Böröczky et al. [START_REF] Böröczky | The mean width of random polytopes circumscribed around a convex body[END_REF], among others.

We in turn transform A z (K) to a subset of R d-1 × R via scaling transforms T λ,z , λ ≥ 1. These transforms yield re-scaled k-face functionals ξ λ,z on the Poisson points T λ,z (P λ ∩ A z (K)), ones which are well approximated by re-scaled k-face functionals on the image under T λ,z of Poisson points in the osculating ball at z. In the large λ limit the latter in turn converge to the scaling limit functionals ξ (∞) given in Definition 2.2.

In this way the expectation and variance asymptotics for k-face functionals on Poisson points in K are obtained by averaging, with respect to all z ∈ ∂K, the respective asymptotics for the re-scaled k-face functionals on Poisson points in osculating balls at z. The limit theory of the latter is established in [START_REF] Calka | Brownian limits, local limits, and variance asymptotics for convex hulls in the ball[END_REF][START_REF] Schreiber | Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points[END_REF] and we shall draw upon it in our approach. For z ∈ ∂K, consider the affine transformation A z which preserves z, the Lebesgue measure, the unit inner normal to z, and which transforms the Weingarten operator at z into r -1 z I d-1 where I d-1 is the identity matrix of R d-1 . Under the action of A z , the number of k-faces of the random convex hull inside the mother body K is preserved. Additionally, ξ k , k ∈ {0, 1, ..., d -1} is stable under the action of A z , namely

ξ k (x, P λ ∩ K) = ξ k (A z (x), A z (P λ ∩ K)). (3.1)
Indeed, A z sends any k-face of K λ to a k-face of A z (K λ ). This follows since affine transformations preserve convexity and convex hulls. A k-face F k of K λ is a.s. the convex hull of (k +1) points from P λ , so it is sent to the convex hull of the images by A z . Moreover, any support hyperplane H such that H ∩ K λ = F k is sent to a support hyperplane of the image of K λ such that its intersection with it is the image of the face F k . So the image of F k is also a k-face of the image of K λ .

Put K z := A z (K). By construction the principal curvatures at z all equal r -1 z . We recall that A z preserves the distribution of P λ so in the sequel, we will make a small abuse of notation by identifying P λ and K λ with A z (P λ ) and A z (K λ ), respectively. Define the osculating ball at z ∈ ∂K to be the ball whose center z 0 := z 0 (z) is at distance r z from z along the inner normal to z. Lemma 4.4 shows that the boundary of the osculating ball B rz (z 0 ) is not far from ∂K z , justifying the terminology.

Given z ∈ ∂K, define f : S d-1 → R + to be the function such that for all u ∈ S d-1 , (z 0 + f (u)u)
is the point of the half line (z 0 + R + u) contained in ∂K z and furthest from z 0 . Thus ∂K z is given by (f (u), u), u ∈ S d-1 . Given z ∈ ∂K we let the inner unit normal be k z := (z 0z)/|zz 0 |. Here and elsewhere we let |w| denote the Euclidean norm of w. For each fixed z ∈ ∂K, we parameterize points w in R d-1 × R by (r, u) where r := |wz 0 | and where u ∈ S d-1 . Henceforth, points (r, u) are with reference to z. For z = (r z , u z ) ∈ ∂K let T z ∼ R d-1 denote the tangent space to S d-1 at u z . The exponential map on the sphere exp d-1 : T z → S d-1 maps a vector v of the tangent space to the point u ∈ S d-1 such that u lies at the end of the geodesic of length |v| starting at z and having direction v. We let the origin of the tangent space be at u z .

3.2. Scaling transformations T λ,z , z ∈ ∂K, λ ≥ 1. Having transformed K to K z , we now re-scale K z for all λ ≥ 1 with a scaling transform denoted T λ,z . Our choice of T λ,z is motivated by the following desiderata. First, consider the epigraph of s λ : S d-1 → R defined by

s λ (u, P λ ) = r z -h K λ (u), u ∈ S d-1 ,
where we recall that r z is the Gaussian curvature radius at z and h

K λ (u) := sup{ x, u , x ∈ K λ } denotes the support function of K λ . Noting that h K λ (u) = sup x∈P λ h x (u) for all u ∈ S d-1 , it
follows that the considered epigraph is the union of epigraphs, which, locally near the vertices of K λ , are of parabolic structure. Thus any scaling transform should preserve this structure, as should the scaling limit. Second, a subset of K z close to z and having a unit volume scaling image should host on average Θ(1) points of P λ,z , that is to say the intensity density of the re-scaled points should be of order Θ [START_REF] Bárány | Random polytopes in smooth convex bodies Mathematika[END_REF]. As in Section 2 of [START_REF] Calka | Brownian limits, local limits, and variance asymptotics for convex hulls in the ball[END_REF], it follows that the transform T λ,z should re-scale K z in the (d -1) tangential directions with factor λ 1/(d+1) and in the radial direction with factor λ 2/(d+1) . It is easily checked that the following choice of T λ,z meet these criteria; cf. Lemma 3.1 below. Throughout we put

β := 1 d + 1
.

Define for all z ∈ ∂K and λ ≥ 1 the finite-size scaling transformation T λ,z : R

+ × S d-1 → R d-1 × R by T λ,z ((r, u)) := (r d z λ) β exp -1 d-1 (u), (r d z λ) 2β (1 - r r z ) := (v ′ , h ′ ) := w ′ . (3.2)
Here exp -1 d-1 (•) is the inverse exponential map, which is well defined on S d-1 \ {-u z } and which takes values in the ball of radius π and centered at the origin of the tangent space T z . We shall

write v ′ := (r d z λ) β exp -1 d-1 u := (r d z λ) β v, where v ∈ R d-1 . We put T λ,z (K z ) := K λ,z ; T λ,z (B rz (z 0 )) := B λ,z ; T λ,z (P λ ∩ K z ) := P λ,z ; T λ,z (P λ ∩ B rz (z 0 )) := P λ,z rz .
We also have the a.e. equality B λ,z = (r

d z λ) β B d-1 (π) × [0, (r d z λ) 2β ), where B d-1 (π) is the closure of the injectivity region of exp d-1 .
We next use the scaling transformations T λ,z on A z (K) to define re-scaled k-face functionals ξ λ,z on re-scaled point sets T λ,z (P λ ∩ K z ); in the sequel we show that these re-scaled functionals converge to the scaling limit functional ξ (∞) given in Definition 2.2. In the special case that K is a ball, we remark that A z (K) = K for all z ∈ ∂K and that T λ,z coincide for all z ∈ ∂K, putting us in the set-up of [START_REF] Calka | Brownian limits, local limits, and variance asymptotics for convex hulls in the ball[END_REF]. defines re-scaled k-face functionals ξ λ,z (w ′ , X ) defined for w ′ ∈ K λ,z and X ⊂ R d by

ξ λ,z (w ′ , X ) := ξ([T λ,z ] -1 (w ′ ), [T λ,z ] -1 (X ∩ K λ,z )). (3.3)
It follows for all z ∈ ∂K, λ ∈ [1, ∞), and x ∈ K z that ξ(x, P λ ∩ K z ) := ξ λ,z (T λ,z (x), P λ,z ).

We shall establish properties of the re-scaled k-face functionals in the next section. For now, we record the distributional limit of the re-scaled point processes P λ,z rz as λ → ∞. Proof. This proof is a consequence of the discussion around (2.14) of [START_REF] Calka | Brownian limits, local limits, and variance asymptotics for convex hulls in the ball[END_REF], but for the sake of completeness we include the details. We find the image by T λ,z of the measure on B rz (z 0 ) given by λr d-1 drdσ d-1 (u). Under T λ,z we have

h ′ := (r d z λ) 2β (1 -r rz ), whence r = r z (1 -(r d z λ) -2β h ′ ). Likewise we have v ′ := (r d z λ) β v, whence v = (r d z λ) -β v ′ . Under T λ,z , the measure r d-1 dr becomes r d-1 dr = (r z (1 -(r d z λ) -2β )) d-1 r 1-2βd z λ -2β dh ′
and dσ d-1 (u) transforms to

dσ d-1 (u) = sin d-2 ((r d z λ) -β |v ′ |) |(r d z λ) -β v ′ | d-2 (r d z λ) -1+2β dv ′
as in (2.17) of [START_REF] Calka | Brownian limits, local limits, and variance asymptotics for convex hulls in the ball[END_REF]. Therefore the product measure λr d-1 drdσ d-1 (u) transforms to

(1 -(r d z λ) -2β h ′ ) d-1 sin d-2 (λ -β |v ′ |) |λ -β v ′ | d-2 dh ′ dv ′ . (3.4)
The total variation distance between Poisson measures is upper bounded by a multiple of the L 1 distance between their densities (Theorem 3.2.2 in [START_REF] Reiss | A course on point processes[END_REF]) and since (1

-(r d z λ) -2β ) (d-1) → 1 as λ → ∞, the result follows.
4 Properties of the re-scaled k-face functional ξ λ,z 4.1. Localization of ξ λ,z . We appeal to results of Reitzner [START_REF] Reitzner | Central limit theorems for random polytopes[END_REF] to show that the re-scaled functionals ξ λ,z given at (3.3) 'localize', that is they are with high probability determined by 'nearby' point configurations.

For all s > 0 consider the inner parallel set of ∂K, namely

K(s) := {x ∈ K : δ H (x, ∂K) ≤ s}, (4.1)
with δ H being the Hausdorff distance. Put

ǫ λ := ( 12d log λ d 3 λ ) β , (4.2)
where d 3 is as in Lemma 5 of Reitzner [START_REF] Reitzner | Central limit theorems for random polytopes[END_REF]. Let B r (x) denote the Euclidean ball of radius r centered at x. We begin with two localization properties of the score ξ. Here and elsewhere we shorthand ξ k by ξ. ), for all z ∈ ∂K, ρ ≥ 1, we have

ξ(x, P λ ∩ K z ) =      ξ(x, P λ ∩ K z (ρǫ 2 λ )) if x ∈ K z (ǫ 2 λ ) 0 if x ∈ K z \ K z (ǫ 2 λ ). ( 4

.3)

(b) There is a constant D 1 such that for all z ∈ ∂K, and x ∈ K z (ǫ 2 λ ) we have

P [ξ(x, P λ ∩ K z ) = ξ(x, P λ ∩ K z ∩ B D1ǫ λ (x))] = O(λ -4d ).
Proof. We prove part (a) with ρ = 1. The proof for ρ > 1 is identical. Let X i , i ≥ 1, be i.i.d.

uniform on K z . For every integer l, let A l be the event that the boundary of co(X 1 , ..., X l ) is contained in K z (ǫ 2 l ). Following nearly verbatim the discussion on page 492 of [START_REF] Reitzner | Central limit theorems for random polytopes[END_REF], we note that P [A c l ] equals the probability that at least one facet of co(X 1 , ..., X l ) contains a point distant at least ǫ 2 l from the boundary of K z , i.e., this is the probability that the hyperplane which is the affine hull of this facet cuts off from K z a cap of height ǫ 2 l which contains no point from X 1 , ..., X l . By Lemma 5 of [START_REF] Reitzner | Central limit theorems for random polytopes[END_REF], the volume of this cap is bounded by d 3 e d+1 l = 12d log l/l. Thus when l is large enough so that (ld)/l > 1/2 (ie. l > 2d) and (12d log l)/l < 1, and using log(1x) < -x, 0 < x < 1, we get

P [A c l ] ≤ l d 1 - 12d log l l l-d < l d 1 d! exp (l -d)(- 12d log l l ) ≤ l d d! l -6d = l -5d d! . ( 4

.4)

Let A λ be the event that the boundary of co(P λ ∩ K) is contained in K z (ǫ 2 λ ). Letting N (λ) be a Poisson random variable with parameter λ we compute

P [A c λ ] = ∞ l=0 P [A c l , N (λ) = l] < |l-λ|≤λ 3/4 P [A c l ] + P [|N (λ) -λ| ≥ λ 3/4 ].
The last term decays exponentially with λ and so exhibits growth O(λ -4d ). By (4.4), the first term has the same growth bounds since

|l-λ|≤λ 3/4 P [A c l ] ≤ 2λ 3/4 max |λ-l|≤λ 3/4 P [A c l ] ≤ 2λ 3/4 1 d! (λ -λ 3/4 ) -5d = O(λ -4d ),
concluding the proof of (a).

We prove assertion (b). By part (a), it suffices to show there is ρ 0 ≥ 1 such that for x ∈ K z (ǫ 2 λ )

P [ξ(x, P λ ∩ K z (ρ 0 ǫ 2 λ )) = ξ(x, P λ ∩ K z (ρ 0 ǫ 2 λ ) ∩ B D1ǫ λ (x))] = O(λ -4d ).
We consider the localization results described on pages 499-502 of [START_REF] Reitzner | Central limit theorems for random polytopes[END_REF] and in the Appendix of [START_REF] Reitzner | Central limit theorems for random polytopes[END_REF]. Using the set-up of Lemma 6 of [START_REF] Reitzner | Central limit theorems for random polytopes[END_REF], we choose m := m(λ) := ⌊(d 6 λ/(4d + 1) log λ) (d-1)β ⌋ points y 1 , ..., y m on ∂K z (here d 6 is the constant of [START_REF] Reitzner | Central limit theorems for random polytopes[END_REF]) such that the Voronoi cells C Vor (y j ), 1 ≤ j ≤ m, partition K z , and such that the diameter of C Vor (y j ) ∩ ∂K z is O(ǫ λ ). Moreover, because all y j are on ∂K z , any bisecting line between two y j makes an angle with ∂K z which is bounded from below. Consequently, since the 'width' of

K z (ǫ 2 λ ) is O(ǫ 2 λ )
, it follows that the diameter of the truncated cells C Vor (y j ) ∩ K z (ǫ 2 λ ) is also O(ǫ λ ). Choose ρ 0 large enough so that K z (ρ 0 ǫ 2 λ ) contains the caps C i , 1 ≤ i ≤ m, given near the end of page 498 of [START_REF] Reitzner | Central limit theorems for random polytopes[END_REF].

For all 1 ≤ j ≤ m, let

S j := {k ∈ {1, 2, ..., m} : C Vor (y k ) ∩ C(y j , d 10 m -2β ) = ∅}
where C(y, h) denotes a cap at y of height h, and where d 10 denote the constant in [START_REF] Reitzner | Central limit theorems for random polytopes[END_REF]. Pages 498-500 of [START_REF] Reitzner | Central limit theorems for random polytopes[END_REF] show the existence of a set A m such that P [A m ] ≥ 1c 16 λ -4d , and on A m the score ξ(x,

P λ ∩ K z (ρ 0 ǫ 2 λ )) at x ∈ K z (ǫ 2 λ ) ∩ C Vor (y j
) is determined by the Poisson points belonging to

U j := U j (x) := k∈Sj C Vor (y k ) ∩ K z (ǫ 2 λ ), (4.5)
where j := j(x) ∈ {1, ..., m} is such that C Vor (y j ) contains x. (Actually [START_REF] Reitzner | Central limit theorems for random polytopes[END_REF] shows this for the score ξ(x, P λ ∩ K z ) and not for ξ(x, P λ ∩ K z (ρ 0 ǫ 2 λ )), but the proof is the same, since ρ 0 is chosen so that K z (ρ 0 ǫ 2 λ ) contains the caps C i , 1 ≤ i ≤ m.) By Lemma 7 of [START_REF] Reitzner | Central limit theorems for random polytopes[END_REF], the cardinality of S j is at most d 8 (d

1/2 10 m -β m β + 1) d+1 = O(1), uniformly in 1 ≤ j ≤ m. This implies that on A m , the score ξ(x, P λ ∩ K z (ρǫ 2 λ )) at x ∈ K z (ǫ 2 λ ) ∩ C Vor (y j
) is determined by the Poisson points in U j , whose diameter is bounded by a constant multiple of the diameter of the truncated cells The next lemma shows localization properties of ξ λ,z . We first require more terminology.

C Vor (y k ) ∩ K z (ρǫ 2 λ ), k ∈ S j ,
Definition 4.1 For all z ∈ ∂K, we put

S λ,z := T λ,z (K z (ǫ 2 λ ) ∩ B 2D1ǫ λ (z)). Note that if w ′ = (v ′ , h ′ ) ∈ S λ,z , then |v ′ | ≤ D 2 (log λ) β for some D 2 not depending on z (here we use sup z∈∂K r z ≤ C). Also, define D 3 by the relation 2[sup z∈∂K r dβ z ]D 1 λ β ǫ λ = D 3 (log λ) β . For all L > 0 and v ∈ R d-1 , denote by C L (v) the cylinder {(v ′ , h) ∈ R d-1 × R : |v ′ -v| ≤ L}.
Due to the non-linearity of T λ,z , localization properties for ξ do not in general imply localization properties for ξ λ,z (w ′ , P λ,z ). However, the next lemma says that if the inverse image of w ′ is close to z, then ξ λ,z (w ′ , P λ,z ) suitably localizes. Lemma 4.2 Uniformly in z ∈ ∂K and w ′ := (v ′ , h ′ ) ∈ S λ,z we have

P [ξ λ,z (w ′ , P λ,z ) = ξ λ,z (w ′ , P λ,z ∩ C D3(log λ) β (v ′ ))] = O(λ -4d ).
Remark. When K is the unit ball we show in [START_REF] Calka | Brownian limits, local limits, and variance asymptotics for convex hulls in the ball[END_REF] that the scores ξ λ,z localize in the following stronger sense: for all w ′ := (v ′ , h ′ ) ∈ K λ,z , there is an a.s. finite random variable R := R(w ′ , P λ,z ) such that ξ λ,z (w ′ , P λ,z ) = ξ λ,z (w

′ , P λ,z ∩ C r (v ′ )) (4.6)
for all r ≥ R, with sup λ P [R > t] → 0 as t → ∞. We are unable to show this latter property for arbitrary smooth K.

Proof. Fix the reference boundary point z ∈ ∂K. Let ρ 0 be as in the proof of Lemma 4.1(b).

For any

A ⊂ R + × R d-1
, we let T λ,z (A) := A λ,z . In view of Lemma 4.1(b), it suffices to show for

w ′ := (v ′ , h ′ ) ∈ S λ,z that P [ξ λ,z (w ′ , (P λ ∩ K z (ρ 0 ǫ 2 λ )) λ,z ) = ξ λ,z (w ′ , (P λ ∩ K z (ρ 0 ǫ 2 λ )) λ,z ∩ C D3(log λ) β (v ′ ))] = O(λ -4d ).
Given w ′ , find j := j(w ′ ) such that C Vor (y j ) contains [T λ,z ] -1 (w ′ ) := x. Recall the definition of U j := U j (x) at (4.5) and recall that the proof of Lemma 4.1 shows that diam(U j ) ≤ D 1 ǫ λ . By the C 3 assumption, if λ is large then for all z ∈ ∂K the projection of U j onto the osculating sphere at z has a diameter comparable to that of U j , i.e., is generously bounded by 2D 1 ǫ(λ). Thus the

spatial diameter of T λ,z (U j ) is bounded by 2[sup z∈∂K r dβ z ]λ β D 1 ǫ λ = D 3 (log λ) β
, by definition of D 3 . In other words

T λ,z (U j ) ⊂ C D3(log λ) β (v ′ ).
(4.7)

However, as seen in the proof of Lemma 4.1, with probability at least 1c 16 λ -4d , the score

ξ λ,z (w ′ , (P λ ∩ K z (ρ 0 ǫ 2 λ )) λ,z
) is determined by the points (P λ ∩ K z (ρ 0 ǫ 2 λ )) λ,z in T λ,z (U j ). In view of (4.7), the proof is complete. Proof. The bound (4.8) follows as in Lemma 7.1 of [START_REF] Calka | Brownian limits, local limits, and variance asymptotics for convex hulls in the ball[END_REF]. To prove (4.9), we argue as follows. Given z ∈ ∂K and w ′ ∈ S λ,z , we let

E := E z (w ′ ) := {ξ λ,z (w ′ , P λ,z ) = ξ λ,z (w ′ , P λ,z ∩ C D3(log λ) β (v ′ ) ∩ (K z (ǫ 2 λ )) λ,z )}.
By Lemmas 4.1(a) and 4.2 we have

P [E c ] = O(λ -4d ).
Let N (s) be a Poisson random variable with parameter s. The cardinality of the point set

P λ,z ∩ C D3(log λ) β (v ′ ) ∩ (K z (ǫ 2 λ )) λ,z , is stochastically bounded by N (C(log λ) β(d-1) • (log λ) 2β ) = N (C log λ),
where C is a generic constant whose value may change from line to line. On the event E the number of k-faces containing

w ′ is generously bounded by N (C log λ) k ≤ (N (C log λ)) k .
We now compute for p ∈ [START_REF] Bárány | Random polytopes in smooth convex bodies Mathematika[END_REF][START_REF] Bárány | The variance of random polytopes[END_REF]:

||ξ λ,z (w ′ , P λ,z )|| p ≤ ||ξ λ,z (w ′ , P λ,z )1(E)|| p + ||ξ λ,z (w ′ , P λ,z )1(E c )|| p .
The first term is bounded by (k+1

) -1 ||N k (C log λ)|| p ≤ M (p)(log λ) k . The second term is bounded by 1 k + 1 card(P λ ) k pr λ -4d/pq , 1/r + 1/q = 1.
We have || card(P λ ) k || pr = O(λ k ) and for p ∈ [START_REF] Bárány | Random polytopes in smooth convex bodies Mathematika[END_REF][START_REF] Bárány | The variance of random polytopes[END_REF] we may choose q sufficiently close to 1 such that λ -4d/pq = O(λ -k ). This gives (4.9).

Remarks. (i) Straightforward modifications of the proof of Lemma 4.1 show that the O(λ -4d ) bounds of that lemma may be replaced by O(λ -md ) bounds, m an arbitrary integer, provided that ǫ λ given at (4.2) is increased by a scalar multiple of m. In this way one could show that Lemma 4.3 holds for moments of any order p > 0. Since we do not require more than fourth moments for ξ λ,z , we do not strive for this generality.

(ii) We do not claim that the bounds of Lemma 4.1 are optimal. By McClullen's bound [START_REF] Mcmullen | The maximum number of faces of a convex polytope[END_REF], the k face functional on an n point set is bounded by Cn d/2 and using this bound for k > d/2 shows that the (log λ) k term in (4.9) can be improved to (log λ) d/2 . The log λ factors could possibly be dispensed with altogether, as mentioned in Section 5.3.

4.3.

Comparison of scores for points in a ball and on K z . The k-face functional of Definition 2.1 on Poisson input on the ball is well understood [START_REF] Calka | Brownian limits, local limits, and variance asymptotics for convex hulls in the ball[END_REF]. To exploit this we need to show that the re-scaled functional ξ λ,z on P λ,z is well approximated by its value on P λ,z rz . We shall also need to show that the pair correlation function for ξ λ,z on P λ,z is well approximated by the pair correlation function for ξ λ,z on P λ,z rz . These approximations are established in the next four lemmas. Our first lemma records a simple geometric fact. Locally around z, the osculating ball to K z may lie inside or outside K z , but it is not far from ∂K z . The next lemma shows that the distance decays like the cube of |v ′ |. Lemma 4.4 For all z ∈ ∂K and v := (r d z λ) -β v ′ we have

r 2βd z λ 2β 1 - f (exp d-1 (v)) r z ≤ D 4 r -1-βd z λ -β |v ′ | 3 . (4.10)
Proof. We first show (4.10) when d = 2. The boundary of the osculating circle at z coincides with ∂K up to at least second order, giving f (0) = r z , f ′ (0) = f ′′ (0) = 0. The Taylor expansion for f It is enough to consider the section of the osculating ball and K z with the plane generated by k z and w. Indeed, we obtain in that plane a two-dimensional mother body with an osculating radius equal to r z at the point z. We may apply the case d = 2 to deduce the required result. Lemma 4.5 Uniformly for z ∈ ∂K and w ′ ∈ S λ,z ∩ B λ,z , we have E ξ λ,z (w ′ , P λ,z )ξ λ,z (w ′ , P λ,z rz ) = O λ -β/2 (log λ) k+(β+1)/2 . (4.11)

around 0 gives |1 -f (v) rz | ≤ 1 6 ||f ′′′ || ∞ r -1 z |v| 3 ,
Proof. For w ′ ∈ S λ,z ∩ B λ,z , we put

E := E(w ′ ) := {ξ λ,z (w ′ , P λ,z ) = ξ λ,z (w ′ , P λ,z ∩ C D3(log λ) β (w ′ ))} (4.12) ∪ {ξ λ,z (w ′ , P λ,z rz ) = ξ λ,z (w ′ , P λ,z rz ∩ C D3(log λ) β (w ′ ))}, so that P [E c ] = O(λ -4d ) by Lemma 4.2. Put F λ,z (w ′ ) := ξ λ,z (w ′ , P λ,z ) -ξ λ,z (w ′ , P λ,z rz ).
By Lemma 4.3 with p = 2, we have

||F λ,z (w ′ )|| 2 ≤ 2M (2)(log λ) k , uniformly in w ′ , λ and z. Recall w ′ := (v ′ , h ′ ). For all w ′ ∈ S λ,z ∩ B λ,z put R(w ′ ) := {(v ′′ , h ′′ ) ∈ R d-1 × R : |v ′′ -v ′ | ≤ D 3 (log λ) β , |h ′′ | ≤ (r d z λ) 2β |1 -r -1 z f (exp d-1 ((r d z λ) -β v ′′ ))|}. (4.13) Write E |F λ,z (w ′ )| = E |(F λ,z (w ′ ))(1(E) + 1(E c ))|.
On E we have F λ,z (w ′ ) = 0, unless the realization of P λ,z puts points in the set R(w ′ ). By the Cauchy-Schwarz inequality and Lemma 4.3 with p = 2 there, we have

E |(F λ,z (w ′ ))1(E)| ≤ 2M (2)(log λ) k (P [1(P λ,z ∩ R(v ′ ) = ∅)]) 1/2 . (4.14)
The Lebesgue measure of R(w ′ ) is bounded by the product of the area of its 'base', that is

(2D 3 (log λ) β ) d-1
and its 'height', which by Lemma 4.4 is at most

D 4 r -1-βd z λ -β |v ′ | + D 3 (log λ) β 3 .
By (3.4), the P λ,z intensity measure of R(w ′ ), denoted by |R(w ′ )|, thus satisfies

|R(w ′ )| ≤ (2D 3 (log λ) β ) d-1 D 4 r -1-βd z λ -β (|v ′ | + D 3 (log λ) β ) 3 . (4.15)
Since 1e -x ≤ x holds for all x it follows that

P [1(P λ,z ∩ R(v ′ ) = ∅)] = 1 -exp(-|R(w ′ )|) ≤ |R(w ′ )|. ( 4
| ≤ D 2 (log λ) β , shows that E |(F λ,z (w ′ ))1(E)| is
bounded by the right hand side of (4.11).

Similarly, Lemma 4.3, the bound

P [E c ] = O(λ -4d
), and the Cauchy-Schwarz inequality give

E |(F λ,z (w ′ ))1(E c )| = O((log λ) k λ -2d
), which is dominated by the right hand side of (4.11). Thus (4.11) holds as claimed.

The next lemma is the analog of Lemma 7.2 in [START_REF] Calka | Brownian limits, local limits, and variance asymptotics for convex hulls in the ball[END_REF]. It justifies the use of the scaling limit terminology for ξ (∞) , as given by Definition 2.2.

Lemma 4.6 For all z ∈ ∂K and (0, h) ∈ K λ,z we have

lim λ→∞ |E ξ λ,z ((0, h), P λ,z ) -E ξ (∞) ((0, h), P)| = 0. Proof. We bound |E ξ λ,z ((0, h), P λ,z ) -E ξ (∞) ((0, h), P)| by |E ξ λ,z ((0, h), P λ,z ) -E ξ λ,z ((0, h), P λ,z rz )| + |E ξ λ,z ((0, h), P λ,z rz ) -E ξ (∞) ((0, h), P)|.
The first term goes to zero by Lemma 4.5 with w ′ = (0, h) and the second term goes to zero by Lemma 7.2 of [START_REF] Calka | Brownian limits, local limits, and variance asymptotics for convex hulls in the ball[END_REF].

We next recall the definition of the pair correlation function for the score ξ as well as for its re-scaled version.

) := E ξ(x, Ξ ∪ y)ξ(y, Ξ ∪ x) -E ξ(x, Ξ)E ξ(y, Ξ). (4.17)
For all λ ≥ 1, z ∈ ∂K, (0, h) ∈ K λ,z , and (v ′ , h ′ ) ∈ K λ,z , define the re-scaled pair correlation function of the k-face functional as

c λ,z ((0, h), (v ′ , h ′ ); P λ,z ) := E ξ λ,z ((0, h), P λ,z ∪ (v ′ , h ′ ))ξ λ,z ((v ′ , h ′ ), P λ,z ∪ (0, h)) -E ξ λ,z ((0, h), P λ,z )E ξ λ,z ((v ′ , h ′ ), P λ,z ). (4.18)
The next lemma shows that the pair correlation function for ξ λ,z on P λ,z is well approximated by the pair correlation function for ξ λ,z on P λ,z rz .

Lemma 4.7 Uniformly for z ∈ ∂K, w 0

′ := (0, h) ∈ S λ,z ∩ B λ,z and w ′ := (v ′ , h ′ ) ∈ S λ,z ∩ B λ,z , we have |c λ,z (w 0 ′ , w ′ ; P λ,z ) -c λ,z (w 0 ′ , w ′ ; P λ,z rz )| = O λ -β/3 (log λ) 2k+β(d+2)/3 . (4.19)
Proof. It suffices to modify the proof of Lemma 4.5. Put

F := E(w 0 ′ ) ∩ E(w ′ )
, where E(w 0 ′ ) and E(w ′ ) are defined at (4.12). We have

P [F c ] = O(λ -4d ) by Lemma 4.2. Write E ξ λ,z (w 0 ′ , P λ,z ∪ (v ′ , h ′ ))ξ λ,z (w ′ , P λ,z ∪ w 0 ′ ) -E ξ λ,z (w 0 ′ , P λ,z rz ∪ w ′ )ξ λ,z (w ′ , P λ,z rz ∪ w 0 ′ ) = E [{ξ λ,z (w 0 ′ , P λ,z ∪ w ′ )ξ λ,z (w ′ , P λ,z ∪ w 0 ′ ) -ξ λ,z (w 0 ′ , P λ,z rz ∪ w ′ )ξ λ,z (w ′ , P λ,z rz ∪ w 0 ′ )}1(F )] + + E [{ξ λ,z (w 0 ′ , P λ,z ∪w ′ )ξ λ,z (w ′ , P λ,z ∪w 0 ′ )-ξ λ,z (w 0 ′ , P λ,z rz ∪w ′ )ξ λ,z (w ′ , P λ,z rz ∪w 0 ′ )}1(F c )] (4.20)
:= I 1 + I 2 .
The random variable in the expectation I 1 vanishes, except on the event

H(w 0 ′ , w ′ ) := {P λ,z ∩ R(w ′ ) = ∅} ∪ {P λ,z ∩ R(w 0 ′ ) = ∅},
where R(w ′ ) and R(w 0 ′ ) are at (4.13). The Hölder inequality

||U V W || 1 ≤ ||U || 3 ||V || 3 ||W || 3 for
random variables U, V, W and Lemma 4.3 imply that

I 1 ≤ 2(M (3)) 2 (log λ) 2k (P [H(w 0 ′ , w ′ )]) 1/3 ,
that is to say

I 1 = O (log λ) 2k r -1-βd z λ -β (log λ) β(d-1) [(|v ′ | + D 3 (log λ) β ) 3 + (D 3 (log λ) β ) 3 ] 1/3
, which for |v ′ | ≤ D 2 (log λ) β satisfies the growth bounds on the right hand side of (4.19).

Now term I 2 in (4.20) is bounded by 2(M (3)) 2 (log λ) 2k (P [F c ]) 1/3
, which is of smaller order than the right hand side of (4.19). This shows that (4.20) also satisfies the growth bounds on the right hand side of (4.19).

It remains to bound

|E ξ λ,z (w 0 ′ , P λ,z )E ξ λ,z (w ′ , P λ,z ) -E ξ λ,z (w 0 ′ , P λ,z rz )E ξ λ,z (w ′ , P λ,z rz )|. (4.21)
Notice that the difference (4.21) differs from

|E ξ λ,z (w 0 ′ , P λ,z )1(F )E ξ λ,z (w ′ , P λ,z )1(F ) -E ξ λ,z (w 0 ′ , P λ,z rz )1(F )E ξ λ,z (w ′ , P λ,z rz )1(F )| (4.22) by at most 4(M (3)) 2 (log λ) 2k (P [F c ]) 1/3 ≤ C(M (3)) 2 (log λ) 2k λ -4d/3 , (4.23)
which is of smaller order than the right hand side of (4. [START_REF] Schreiber | Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points[END_REF]).

Now we control the difference (4.22) which we write as |E e 1 E e 2 -E e 3 E e 4 |, where e 1 := ξ λ,z (w 0 ′ , P λ,z )1(F ), e 2 := ξ λ,z (w ′ , P λ,z )1(F ), e 3 := ξ λ,z (w 0 ′ , P λ,z rz )1(F ), and e 4 := ξ λ,z (w ′ , P λ,z rz )1(F ). The proof of Lemma 4.5 (with E replaced by F ) shows that

E |e 1 -e 3 | = O(λ -β/2 (log λ) k+(β+1)/2 ) (4.
O(λ -β/2 (log λ) 2k+(β+1)/2 ) + O((log λ) 2k λ -4d/3 ), (4.26)
i.e., is bounded by the right-hand side of (4.19).

Our last lemma describes a decay rate for c(x, y; P λ ∩ K z ), a technical fact used in the sequel.

Lemma 4.8 For all z ∈ ∂K and x, y

∈ K z (ǫ 2 λ ) with |x -y| ≥ 2D 1 ǫ λ , we have lim λ→∞ λ 1+2β c(x, y; P λ ∩ K z ) = 0. Proof. Fix x ∈ K z (ǫ 2 λ ).
To lighten the notation we abbreviate P λ ∩ K z by P λ in this proof only. For y ∈ K z (ǫ 2 λ ), put If |x -y| ≥ 2D 1 ǫ λ , then ξ(x, P λ ∪ y) and ξ(y, P λ ∪ x) are independent on E, giving

E [ξ(x, P λ ∪ y)ξ(y, P λ ∪ x)1(E)] = E [ξ(x, P λ )1(E) • ξ(y, P λ )1(E)] = E [ξ(x, P λ )1(E)] • E [ξ(y, P λ )1(E)].
Writing

1(E) = 1 -1(E c ) gives E [ξ(x, P λ ∪ y)ξ(y, P λ ∪ x)1(E)] = (E ξ(x, P λ ) -E [ξ(x, P λ )1(E c )]) • (E ξ(y, P λ ) -E [ξ(y, P λ )1(E c )]) = E ξ(x, P λ )E ξ(y, P λ ) + G(x, y),
where

G(x, y) := -E ξ(x, P λ )E [ξ(y, P λ )1(E c )] -E ξ(y, P λ )E [ξ(x, P λ )1(E c )] + E [ξ(x, P λ )1(E c )] • E [ξ(y, P λ )1(E c )].
Let N (λ) := card(P λ ∩ K z ). By McClullen's bounds [START_REF] Mcmullen | The maximum number of faces of a convex polytope[END_REF]

)1(E c )]| = O(λ d/2 λ d/2 (P [E c ]) 1/2 ) = o(λ -1-2β ),
where the last estimate easily follows by (4.27). The other two terms comprising G(x, y) have the same asymptotic behavior and so G(x, y) = o(λ -1-2β ).

On the other hand, E [ξ(x, P λ ∪ y)ξ(y, P λ ∪ x)1(E)] differs from E ξ(x, P λ ∪ y))ξ(y, P λ ∪ x) by

E [ξ(x, P λ ∪ y)ξ(y, P λ ∪ x)1(E c )]. The Hölder inequality ||U V W || 1 ≤ ||U || 4 ||V || 4 ||W || 2 shows that this term is o(λ -1-2β ).
Thus E ξ(x, P λ ∪ y))ξ(y, P λ ∪ x) and E ξ(x, P λ )E ξ(y, P λ ) differ from E [ξ(x, P λ ∪ y)ξ(y, P λ ∪

x)1(E)] by o(λ -1-2β ), concluding the proof of Lemma 4.8.

Proof of Theorem 2.1

Recall that M(K) denotes the medial axis of K and, for every z ∈ ∂K the inner unit-normal vector of ∂K at z is k z . Put t m (z) := inf{t > 0 : z + tk z ∈ M(K)}. Then the map ϕ : (z, t) -→ (z + tk z ) is a diffeomorphism from {(z, t) : z ∈ ∂K, 0 < t < t m (z)} to Int(K) \ M(K). In particular, z -→ -k z is the Gauss map and its differential is the shape operator or Weingarten map W z , which we recall has eigenvalues C z,1 , • • • , C z,d-1 . Consequently, the Jacobian of ϕ may be written as det(I -tW z ) = d-1

i=1 (1 -tC z,i ).

5.1. Proof of expectation asymptotics (2.5). Fix g ∈ C(K) and let ξ and µ ξ λ denote a generic k face functional and k face measure, respectively. Recall that we may uniquely write x ∈ K \ M(K) as x := (z, t), where z ∈ ∂K, and t ∈ (0, ∞) is the distance between x and z. Write

λ -1+2β E [ g, µ ξ λ ] = λ 2β K g(x)E ξ(x, P λ ∩ K)dx = λ 2β z∈∂K tm(z) 0 g((z, t))E ξ((z, t), P λ ∩ K) • Π d-1 i=1 (1 -tC z,i )dtdz.
For each z ∈ ∂K, we apply the transformation A z to K. Recalling from (3.1) that ξ is stable under

A z , we have E ξ((z, t), P λ ∩ K) = E ξ((z, t), P λ ∩ K z ), since A z (z, t) := (z, t) and A z (P λ ∩ K)

D = P λ ∩ K z . It follows that λ -1+2β E [ g, µ ξ λ ] = λ 2β z∈∂K tm(z) 0 g((z, t))E ξ((z, t), P λ ∩ K z ) • Π d-1 i=1 (1 -tC z,i )dtdz.
By Lemma 4.1(a), the bound (4.9) with p = 2, and the Cauchy-Schwarz inequality, it follows that uniformly in x ∈ K z \ K z (ǫ 2 λ ) we have lim λ→∞ λ 2β E ξ(x,

P λ ∩ K z ) = 0. Since sup λ≥1 sup x∈Kz\Kz(ǫ 2 λ ) λ 2β E ξ(x, P λ ∩ K z ) ≤ C,
the bounded convergence theorem shows that we can restrict the range of integration of t to the interval [0, ǫ 2 λ ] with error o(1). This gives

λ -1+2β E [ g, µ ξ λ ] = λ 2β z∈∂K ǫ 2 λ 0 g((z, t))E ξ((z, t), P λ ∩ K z ) • Π d-1 i=1 (1 -tC z,i )dtdz + o(1). (5.1)
Changing variables with t = r z (r d z λ) -2β h and using h = (r

d z λ) 2β (r z -r)/r z = (r d z λ) 2β (t/r z ) gives ξ((z, t), P λ ∩ K z ) = ξ λ,z ((0, h), P λ,z ). Letting h(λ, z) := r -1+2βd z λ 2β ǫ 2 λ we get λ -1+2β E [ g, µ ξ λ ] = z∈∂K r 1-2βd z h(λ,z) 0 g((z, o u (1)))E ξ λ,z ((0, h), P λ,z ) • Π d-1 i=1 (1 -o u (1))dhdz + o(1)
where o u (1) denotes a quantity tending to zero as λ → ∞, uniformly in z ∈ ∂K and uniformly in h ∈ [0, h(λ, z)], not necessarily the same at each occurrence.

Note that (0, h) belongs to S λ,z ∩ B λ,z and so we may apply Lemma 4.5 to ξ λ,z ((0, h), P λ,z ).

Thus, with w ′ set to (0, h) in Lemma 4.5, we have

sup z∈∂K sup h∈[0,h(λ,z)] h(λ, z) E ξ λ,z ((0, h), P λ,z ) -E ξ λ,z ((0, h), P λ,z rz ) = o(1),
and so we may replace E ξ λ,z ((0, h), P λ,z ) by E ξ λ,z ((0, h), P λ,z rz ) with error o(1). We also have r 1-2βd z = κ(z) 1/(d+1) . In other words,

λ -1+2β E [ g, µ ξ λ ] = z∈∂K κ(z) 1/(d+1) h(λ,z) 0 g((z, o u (1))E ξ λ,z ((0, h), P λ,z rz ) • Π d-1 i=1 (1 -o u (1))dhdz + o(1)
.

By Lemma 3.2 of [START_REF] Calka | Brownian limits, local limits, and variance asymptotics for convex hulls in the ball[END_REF], the integrand is dominated by an exponentially decaying function of h, uniformly in z and λ.

The continuity of g, and the dominated convergence theorem give

lim λ→∞ λ -1+2β E [ g, µ ξ λ ] = z∈∂K g(z)κ(z) 1/(d+1) ∞ 0 E ξ (∞) ((0, h), P)dh dz. (5.2)
This gives (2.5), as desired.

5.2. Proof of variance asymptotics (2.6). Recalling (4.17), for fixed g ∈ C(K) we have

λ -1+2β Var[ g, µ ξ λ ] = λ 2β K g(x) 2 E ξ 2 (x, P λ ∩ K)dx + + λ 1+2β K K g(x)g(y)c(x, y; P λ ∩ K)dydx := I 1 (λ) + I 2 (λ).
Following the proof of (2.5) until (5.2) shows that

lim λ→∞ I 1 (λ) = z∈∂K g(z) 2 κ(z) 1/(d+1) ∞ 0 E (ξ (∞) ((0, h), P)) 2 dhdz. (5.3)
Turning to I 2 (λ), write x in curvilinear coordinates (z, t) with respect to ∂K. This gives

dx = Π d-1 i=1 (1 -tC z,i )dtdz.
Apply the map A z , write A z (y) = ȳ for y ∈ K, and use stability (3.1) to get

I 2 (λ) = λ 1+2β z∈∂K t ′ m (z) 0 ȳ∈Kz g((z, t))g(ȳ)c((z, t), ȳ; P λ ∩ K z )dȳ • Π d-1 i=1 (1 -tC z,i )dtdz + o(1). (5.4)
Here c((z, t), ȳ;

P λ ∩K z ) = E ξ((z, t), P λ ∩K z ∪ ȳ)ξ(ȳ, P λ ∩K z ∪(z, t))-E ξ((z, t), P λ ∩K z )E ξ(ȳ, P λ ∩K z ).
The McClullen bound [START_REF] Mcmullen | The maximum number of faces of a convex polytope[END_REF] gives |c((z, t), ȳ;

P λ ∩ K z )| ≤ CE [N (λ) d ] ≤ Cλ d , (5.5)
where here N (λ) denotes the cardinality of P λ ∩ K z .

We make the following three modifications to the triple integral (5.4), each one giving an error of o(1): (iii) Replace the integration domain [0, t ′ m (z)] by [0, ǫ 2 λ ], as at (5.1). These modifications yield

(i) Replace the integration domain {ȳ ∈ K z } by {ȳ ∈ K z (ǫ 2 λ )}. Indeed, uniformly in ȳ ∈ K z \ K z (ǫ 2
I 2 (λ) = (5.6) = λ 1+2β z∈∂K ǫ 2 λ 0 ȳ∈Kz(ǫ 2 λ )∩B2D 1 ǫ λ ((z,t))
g((z, t))g(ȳ)c((z, t), ȳ;

P λ ∩K z )dȳ•Π d-1 i=1 (1-tC z,i )dtdz+o(1).
Changing variables with ȳ = (r, u) gives dȳ = r d-1 drdσ d-1 (u) and it also gives

T λ,z ((r, u)) = ((r d z λ) β exp -1 d-1 (u), (r d z λ) 2β (1 - r r z )) = ((r d z λ) β v, h ′ ) = (v ′ , h ′ ) = w ′ .
Thus the covariance c((z, t), ȳ; P λ ∩ K z ) transforms to c λ,z ((0, h), (v ′ , h ′ ); P λ,z ). Now change vari-

ables with t = r z (r d z λ) -2β h, v ′ = (r d z λ) β v, and h ′ = (r d z λ) 2β (1 -r rz ). The differential λ 1+2β Π d-1 i=1 (1 -tC z,i )r d-1 drdσ d-1 (u)dtdz transforms to the differential λ 1+2β Π d-1 i=1 (1-r z (r d z λ) -2β hC z,i )((1-(r d z λ) -2β h ′ )r z ) d-1 r z (r d z λ) -2β dh ′ ×(r d z λ) -β(d-1) dv ′ r z (r d z λ) -2β dhdz = Π d-1 i=1 (1 -r z (r d z λ) -2β hC z,i )(1 -(r d z λ) -2β h ′ ) d-1 r 1-2βd z dh ′ dv ′ dhdz.
The upper limit of integration ǫ 2 λ in (5.6) changes to h(λ, z) and the domain of integration K z (ǫ 2 λ ) ∩ B 2D1ǫ λ ((z, t)) gets mapped to S λ,z . This gives

I 2 (λ) = z∈∂K h(λ,z) 0 (v ′ ,h ′ )∈S λ,z G λ (h ′ , v ′ , h, z)dh ′ dv ′ dhdz + o(1), (5.7)
where, recalling r 1-2βd z = κ(z) 1/(d+1) , we get

G λ (h ′ , v ′ , h, z) := κ(z) 1/(d+1) g((z, o u (1)))g(r z (1 -o u (1)), (r d z λ) -β v ′ ) •c λ,z ((0, h), (v ′ , h ′ ); P λ,z )Π d-1 i=1 (1 -o u (1))(1 -o u (1)) d-1 .
We next restrict the integration domain S λ,z to S λ,z ∩ B λ,z since by Lemma 4.4 and the moment bounds (4.9) we have z∈∂K h(λ,z)

0 (v ′ ,h ′ )∈S λ,z ∩(B λ,z ) c G λ (h ′ , v ′ , h, z)dh ′ dv ′ dhdz = o(1)
.

By Lemma 4.7, uniformly on the range {(v ′ , h ′ ) ∈ S λ,z ∩ B λ,z } and uniformly over h ∈ [0, h(λ, z)],

the covariance term c λ,z ((0, h), (v ′ , h ′ ); P λ,z ) differs from the covariance term c λ,z ((0, h), (v ′ , h ′ ); P λ,z rz ) by a term of order λ -β/3 , modulo logarithmic terms. The integral of this difference over (h ′ , v ′ , h, z) ∈ S λ,z × [0, h(λ, z)] × ∂K is also o(1). This gives

I 2 (λ) = z∈∂K |h|≤h(λ,z) (v ′ ,h ′ )∈S λ,z ∩B λ,z Gλ (h ′ , v ′ , h, z)dh ′ dv ′ dhdz + o(1), (5.8)
where Gλ (h ′ , v ′ , h, z) = κ(z) 1/(d+1) g((z, o u (1))g(r z (1o u (1), o u (1))

•c λ,z ((0, h), (v ′ , h ′ ); P λ,z rz )Π Recalling the definition of ζ ξ (∞) at (2.3) we get via Lemma 7.2 of [START_REF] Calka | Brownian limits, local limits, and variance asymptotics for convex hulls in the ball[END_REF] that lim λ→∞ Gλ (h ′ , v ′ , h, z) = κ(z) 1/(d+1) g(z) 2 ζ ξ (∞) ((0, h), (v ′ , h ′ ); P).

The first part of Lemma 7.3 of [START_REF] Calka | Brownian limits, local limits, and variance asymptotics for convex hulls in the ball[END_REF] shows that c λ,z ((0, h), (v ′ , h ′ ); P λ,z rz ) is dominated by an integrable function of h ′ , v ′ , h and z on [0, ∞) × R bounds, and (iii) show that c λ,z ((0, h), (v ′ , h ′ ); P λ,z ) decays exponentially in |v ′ | and h ′ , showing that G λ (h ′ , v ′ , h, z) is integrable. We could then directly apply the dominated convergence theorem to E ξ λ,z ((0, h), P λ,z ) and c λ,z ((0, h), (v ′ , h ′ ); P λ,z ) without needing the error approximations of Lemmas 4.5 and 4.7.

6 Proof of Theorem 1.2

The image of K by x -→ vol(K) -1/d • x is a convex body of unit volume so without loss of generality, we may assume in this section that vol(K) = 1. The proof of Theorem 1.2 via Theorem 1.1 is a rewriting of a result previously obtained by Vu (see [START_REF] Vu | Central limit theorems for random polytopes in a smooth convex set[END_REF], Proposition 8.1) in the case k = 0.

For sake of completeness, we include here a proof which does not use any large deviation result for f k (K λ ). The method uses a coupling of the Poisson point process of intensity n and the binomial point process.

Let X i , i ≥ 1, be a sequence of i.i.d. uniform random variables in K(ǫ 2 n ) and put X n := {X 1 , • • • , X n }. For sake of simplicity, we denote by f k (X n ∩ K(ǫ 2 n )) the number of k-dimensional faces of the convex hull of X n . In particular, we have

f k (X n ∩ K(ǫ 2 n )) := Xi∈Xn∩K(ǫ 2 n ) ξ k (X i , X n ∩ K(ǫ 2 n )).
We start with two preliminary lemmas which describe the growth of f k (X n ∩ K(ǫ 2 n )).

Lemma 6.1 For all k ∈ {0, 1, ..., d -1} there is a set F (n), P [F (n) c ] = O(n -4d ), and a constant

C 1 ∈ (0, ∞) such that on F (n) |f k (X n ∩ K(ǫ 2 n )) -f k (X n+1 ∩ K(ǫ 2 n ))| ≤ C 1 (log n) k+1 . (6.1)

Theorem 1 . 2

 12 For all k ∈ {0, 1, ..., d -1} we havelim n→∞ n -(d-1)/(d+1) Varf k (K ′ n ) = F k,d (vol(K)) -(d-1)/(d+1)∂K κ(z) 1/(d+1) dz.(1.3)

2. 1 ,

 1 giving expectation and variance asymptotics for the empirical k-face measure, which includes Theorem 1.1 as a special case. Theorem 2.1 also shows that the constants F k,d of Theorem 1.1 may be expressed in terms of integrals of one and two point correlation functions of a scaling limit k-face functional ξ (∞) k associated with parabolic growth processes. Section 3 introduces an affine transform of K and a scaling transform of the affine transform to link the finite volume k face functional with its infinite volume scaling limit counterpart ξ (∞) k . Section 4 contains the main technical aspects of the paper, focussing on the properties of the re-scaled k-face functionals. In particular Lemmas 4.5 and 4.7 show that the one and two point correlation functions of the rescaled k-face functional on the affine transform of K are well approximated by the corresponding one and two point correlation functions of the re-scaled k-face functional on an osculating ball. In this way the expectation and variance asymptotics for f k (K λ ), K an arbitrary smooth body, are controlled by the corresponding asymptotics for f k (K λ ) when K is a ball. The latter asymptotics are established in[START_REF] Calka | Brownian limits, local limits, and variance asymptotics for convex hulls in the ball[END_REF]. Section 5 contains the proof of Theorem 2.1 which implies Theorem 1.1.

3. 1 .

 1 Affine transformations A z , z ∈ K. Let M(K) be the medial axis of K. M(K) has Lebesgue measure zero and we parameterize points x ∈ K \ M(K) by x := (z, t), where z ∈ ∂K is the unique boundary point closest to x and where t ∈ [0, ∞) is the distance between x and z. Denote by C z,1 , • • • , C z,d-1 the principal curvatures of ∂K at z, i.e. the eigenvalues of the Weingarten operator at z. Let κ(z) := d-1 i=1 C z,i be the Gaussian curvature at z, so that the Gaussian curvature radius r z satisfies κ(z) = r -(d-1) z .

3. 3 .

 3 Re-scaled k-face functionals ξ λ,z , z ∈ ∂K, λ ≥ 1. Fix λ ∈ [1, ∞) and z ∈ ∂K. Let ξ := ξ k be a generic k-face functional, as given in Definition 2.1. The inverse transformation [T λ,z ] -1

Lemma 3 . 1

 31 Fix z ∈ ∂K. As λ → ∞, we have P λ,z rz D -→ P in the sense of total variation convergence on compact sets.

Lemma 4 . 1

 41 (a) With probability at least 1 -O(λ -4d

  and is thus determined by points distant at most D 1 ǫ λ from x, D 1 a constant. Since P [A c m ] ≤ c 16 λ -4d , this proves assertion (b).

4. 2 .Lemma 4 . 3

 243 Moment bounds for ξ λ,z . We use the localization results to derive moment bounds for the re-scaled k-face functionals ξ λ,z . For a random variable W and all p > 0, we let||W || p := (E |W | p ) 1/p . Let ξ := ξ k , k ∈ {0, 1, ..., d -1}. For all p ∈ [1, 4] there are constants M (p) := M (p, k) ∈ (0, ∞) such that sup z∈∂K sup λ≥1 sup w ′ ∈B λ,z ||ξ λ,z (w ′ , P λ,z rz )|| p ≤ M (p∈S λ,z||ξ λ,z (w ′ , P λ,z )|| p ≤ M (p)(log λ) k . (4.9)

  whence the result. We now consider the case d ≥ 3. Let exp d-1 (v) := cos(|v|)k z + sin(|v|)w, where w := v/|v|.

Definition 4 . 2 (

 42 Pair correlation functions) For all x, y ∈ K z , any random point set Ξ ⊂ K z , and any ξ we put c(x, y; Ξ) := c ξ (x, y; Ξ

24) and E |e 2 -e 4 |

 24 = O(λ -β/2 (log λ) k+(β+1)/2 ) (4.25) Since |E e 1 E e 2 -E e 3 E e 4 | ≤ |E e 1 ||E e 2 -E e 4 | + |E e 4 ||E e 1 -E e 3 | it follows that (4.21) is bounded by

E

  := E(x, y) := {ξ(x, P λ ) = ξ(x, P λ ∩ B D1ǫ λ (x))} ∪ {ξ(y, P λ ) = ξ(y, P λ ∩ B D1ǫ λ (y))}. Lemma 4.1(b) gives P [E c ] = O(λ -4d ).(4.27)

  λ ) we have lim λ→∞ λ 1+2β c((z, t), ȳ; P λ ∩ K z ) = 0, by Lemma 4.1(a), the bound (5.5), and the Cauchy-Schwarz inequality. Since sup λ≥1 sup (z,t),ȳ∈Kz\Kz(ǫ 2 λ )λ 1+2β c((z, t), ȳ;P λ ∩ K z ) ≤ C,the assertion follow by the bounded convergence theorem.(ii) Replace the integration domain {ȳ ∈ K z (ǫ 2 λ )} by {ȳ ∈ K z (ǫ 2 λ ) ∩ B 2D1ǫ λ ((z, t))} (use Lemma 4.8 and the bounded convergence theorem).

d- 1 i=1 ( 1 -

 11 o u (1))(1o u (1)) d-1 .

0 E 0 R d- 1 ∞ 0 ζ

 0010 d-1 × [0, ∞) × ∂K. Since sup z∈∂K | r d+1 z | and ||g|| ∞ are both bounded and since the integration domain S λ,z ∩B λ,z increases up to R d-1 ×[0, ∞), the dominated convergence theorem gives lim λ→∞I 2 (λ) = z∈∂K g(z) 2 κ(z) 1/(d+1) ∞) ((0, h), (v ′ , h ′ ); P)dh ′ dv ′ dhdz. (ξ (∞) ((0, h), P)) 2 dh + + κ(z) 1/(d+1) ∞ ξ (∞) ((0, h), (v ′ , h ′ ); P)dh ′ dv ′ dhdz.(5.10)Recalling the definition of σ 2 (ξ (∞) ) at (2.4), this yieldslim λ→∞ λ -1+2β Var[ g, µ ξ λ ] = σ 2 (ξ (∞) ) ∂K g(z) 2 κ(z) 1/(d+1) dz. (5.11)This concludes the proof of variance asymptotics and the proof of Theorem 2.1.Remark. If one could show that ξ λ,z localize in the sense of (4.6), then one could show that the moment bounds of Lemma 4.3 are independent of λ. We expect that one could subsequently weaken the C 3 boundary assumption to a C 2 assumption by making these three changes: (i) replace the right-hand side of (4.10) with o(1)|v ′ | 2 , (ii) in Lemmas 4.5 and 4.7, drop the restrictions w ′ 0 , w ′ ∈ S λ,z ∩ B λ,z , and replace the bounds on the right-hand side of (4.11) and (4.19) with o(1)

  for the number of k-dimensional faces and standard moment bounds for Poisson random variables we have ||ξ(x, P λ )|| 1 ≤ C||N d/2 (λ)|| 1 ≤ Cλ d/2 and similarly ||ξ(y, P λ )|| 2 ≤ Cλ d/2 . By the Cauchy-Schwarz inequality, it follows that |E ξ(x, P λ )E [ξ(y, P λ

* Research partially supported by French ANR grant PRESAGE. * * Research supported in part by NSF grant DMS-1106619

Proof. As in the proof of Lemma 4.1 and as on the pages 499-502 of [START_REF] Reitzner | Central limit theorems for random polytopes[END_REF], there is a set F 1 (n) with

), such that on F 1 (n) we have for

Thus on F 1 (n) we have

The proof of Lemma 4.3 shows that for

The same occurs for ξ k (X n+1 , X n+1 ). Now on the set F (n) := F 1 (n) ∩ F 2 (n) ∩ F 3 (n) we get (6.1), concluding the proof of Lemma 6.1. Lemma 6.2 For all k ∈ {0, 1, ..., d-1} there is a constant C 2 such that for all integers l = 1, 2, ..., n we have

Proof. We have

By Lemma 6.1, the ith summand is bounded by

, the result follows.

For every λ > 0, let N (λ) denote a Poisson variable of mean λ and for every integer n and p ∈ (0, 1), let Bi(n, p) denote a Binomial variable of parameters n and p. The next result yields Theorem 1.2.

Theorem 6.1 For all k ∈ {0, 1, ..., d -1} we have

Proof. For all integers m we put

By (1.2), we have

It is thus enough to show

Given the binomial and Poisson distributions L(Bi(n, ǫ 2 n )) and L(N (nǫ 2 n )), there exist coupled random variables Bi(n, ǫ 2 n ) and N (nǫ 2 n ) such that

see e.g. (1.4) and (1.23) of [START_REF] Barbour | Poisson Approximation[END_REF].

Enumerate the points

), consider the coupled point set Y n obtained by discarding or adding i.i.d. points X i in K(ǫ 2 n ):

We use this coupling of the point sets

n ) in all that follows. Denoting the convex hull of m i.i.d. points X 1 , ..., X m on K(s) by K(s) ′ m , we have

where the last equality follows from the O(n -4d ) probability bounds of Lemma 4.1(a), the bounds

, as well as a standard application of the Cauchy-Schwarz inequality. Let

.

By the Bernstein inequality there is a constant C 4 such that for all p ∈ (0, 1/2) we have

with probability at least 1-O(n -4d ). By Proposition A.2.3 of [START_REF] Barbour | Poisson Approximation[END_REF], and taking C 4 larger if necessary, we also have

with probability at least 1 -O(n -4d ). A modification of Lemma 6.2 shows that there is a set G n [START_REF] Bárány | Random polytopes in smooth convex bodies Mathematika[END_REF] with probability at least 1 -O((log n) 1+1/(d+1) n 1/2-1/(d+1)-4d ) such that on G n (1) we have

Similarly, there is a set G n (2) with probability at least 1 -O((log n) 1+1/(d+1) n 1/2-1/(d+1)-4d ) such that on G n (2) we have

By McClullen's bound [START_REF] Mcmullen | The maximum number of faces of a convex polytope[END_REF] |f

always holds. It follows by the Cauchy-Schwarz inequality that

whence in view of (6.5)

It follows that

This shows (6.2) and concludes the proof of Theorem 6.1.


Axel Hutt

Jérémie Lefebvre

André Longtin

Delay stabilizes stochastic systems near an non-oscillatory instability

Keywords: 02, 50, Fz -Stochastic analysis 02, 30, Ks -Delay and functional equations 87, 19, lc -Noise in the nervous system

The work discovers a stochastic bifurcation in delayed systems in the presence of both delay and additive noise. To understand this phenomenon we present a stochastic center manifold method to compute a non-delayed stochastic order parameter equation for a scalar delayed system driven by additive uncorrelated noise. The derived order parameter equation includes additive and multiplicative white and coloured noise. An illustrative neural system with delayed self-excitation reveals stationary states that are postponed by combined additive noise and delay. A final brief analytical treatment of the derived order parameter equation reveals analytically the shift of the stationary states which depends on the delay and the noise strength.

Nonlinear time-delayed systems arise in numerous areas [START_REF] Erneux | Applied Delay-Differential Equations[END_REF]. The effect of noise on such systems is of great interest, for example in the context of delayed laser dynamics [START_REF] Buldu | [END_REF]. Biological systems are rife with delays, and noise is often significant enough to e.g. alter bifurcations in genetic networks [3], even producing stationary states -as maxima of the stationary probability density -that have no deterministic counterpart [4]. The same is true for neural systems involved in the control of e.g. the pupil [5], eye movements [6], balance [7] and neural rhythms [8]. Analytical insights into the properties of stochastic delaydifferential systems (SDDEs) have been hindered by the fact that Fokker-Planck analysis is not valid for such non-Markovian dynamics. Advances have relied on approximations using e.g. small delay expansions [9], multiscale methods [10], delayed random walks [11], as well as coarsegrained dynamics [12] naturally applied to intrinsically bistable delayed systems.

Yet a general theory for SDDEs is still lacking, especially near instabilities. A technique to calculate the effects of noise well into nonlinear regimes would be an important step, and is the focus of our Letter. We highlight delayed dynamics involving a pitchfork bifurcation, but our work is in fact applicable to all systems having a one-dimensional normal form in the absence of delay. We consider the system dx(t) dt = -x(t) + βx(t -τ ) -γx 3 (t -τ ) + κξ(t)

with β, γ ∈ R and the delay τ . Here ξ(t) is Gaussian white noise (GWN) with ξ(t) = 0, ξ(t)ξ(t ) = 2δ(t -t ). The noise strength is denoted by κ, and • denotes the ensemble average. A motivation for our work arises in the context of a single neuron with global excitatory delayed feedback from a population, which exhibits a spatially synchronized mean membrane potential U (t) obeying

dU (t) dt = -αU (t) + K 0 S[U (t -τ 0 )] + E 0 + ξ(t) .( 2
)
The parameter α is the mean decay rate of synaptic responses and K 0 represents the excitatory feedback level. The sigmoid function S(V ) = S 0 /(1 + exp(-c(V -V thr ))) reflects the population firing rate function with constant parameters S 0 , V thr and c and τ 0 denotes the fixed feedback delay. The global external stimulus includes the mean background activity E 0 from external neural processes, and the small input ξ(t) represents random fluctuations around that mean [8]. Equation (1) includes the pre-factors β = S 0 K 0 c/4α > 0, γ = S 0 K 0 c 3 /48α > 0 and the rescaled time and delay t → t/α and τ = ατ 0 , respectively. The pre-factors β and γ are related by γ = ηβ 3 with η = 4(α/S 0 K 0 ) 2 /3. This dependence of linear and nonlinear pre-factors in Eq. ( 1) is a generic property of neural systems. Considering small deviations of U (t) from V thr , i.e. x(t) ≡ U (t) -V thr , |x(t)| V thr , and Taylorexpanding S about V thr up to cubic order, yields the delayed Langevin equation Eq. [START_REF] Erneux | Applied Delay-Differential Equations[END_REF]. The resting states of Eq. ( 1) for zero noise are x 0 = 0, x 1,2 = ± ε/η(ε + 1) 3 with the new control parameter ε = β -1 > 0. We find the well-expected pitchfork bifurcation for small ε. Moreover x 0 is metastable for ε = 0.

Figure 1 shows the stationary probability density P s (x) of the stochastic process (1) for vanishing delay (Fig. 1(a)) and a large delay τ = 2 (Fig. 1(b)). For τ = 0, we observe the expected uni-modal density for ε < 0 and the bi-modal density for ε > 0. In the case of a large delay τ = 2 the bi-modal density emerges for much larger values of ε compared to the case τ = 0. This indicates that the critical instability point is moved to larger values of the control parameter, i.e. the system exhibits stationary activity about the state x = 0 for larger control parameters than expected from a deterministic analysis. Moreover, it is well-known that noise-free delayed scalar systems such as (1) for κ = 0 [13] and stochastic non-delayed scalar systems with additive GWN, i.e. (1) with τ = 0, have identical stationary states, cf. dotted line in Fig. 1. Consequently the stochastic bifurcation shown in Fig. 1(b) occurs only in the presence of both delay and additive noise. Hence introducing delay in a system subjected to additive noise stabilizes the system. Such a noise-induced transition is known only for multiplicative GWN noise in delayed systems [5,14]. Numerical work had also shown that additive GWN can shift a Hopf bifurcation in a scalar SDDE [5], and here we provide analytics that explain a novel and simpler manifestation of this phenomena.

Fig. 1: The stationary probability density (PDF) of the stochastic process (1) computed numerically without (a) and with delay (b). The white dotted line denotes the stationary states of (1) in the absence of both delay and noise. Simulations used a delayed Euler-Maruyama integration method with iteration step dt = 0.01 and parameters η = 80 (used throughout) and κ = 0.015. The PDFs were computed using an ensemble average of 15 • 10 4 trajectories in the stationary state reached at t = 1000.

To investigate this numerical finding analytically, the constant control parameter ε is made a dynamic variable with dε/dt = 0, allowing an easier application of the center manifold approach. This yields the augmented system

dx(t) dt = -x(t) + x(t -τ ) + F (x, ε, t) , dε dt = 0 (3)
with F (x, ε, t) = εx(t-τ )-η(1+ε) 3 x 3 (t-τ )+κξ(t) including both the nonlinear interaction terms and the random input. Moreover, the linear terms no longer depend on ε. Mathematically, the system in Eq. ( 3) has two variables but evolves in an infinite-dimensional space due to the delay. Hence it is practical to introduce the 2-dimensional vector function

z t (θ) = (x(t + θ), ε) T = (z t (θ) 1 , z t (θ) 2 ) T with -τ ≤ θ ≤ 0, where z t lives in the Banach space of continuous maps C B ≡ C([-τ, 0], R × R) [15,16]. Then (3) reads d dt z t (θ) = A(z t (θ)) + X o F [z t , t] , (4)
where

A = ∂z t /∂θ for -τ ≤ θ < 0, A = L[z t ] for θ = 0, with the linear operator L[z t ] = (-z t (0) 1 + z t (-τ ) 1 , 0) T which is re-formulated as functional operator L[z t ] = 0 -τ -δ(θ) + δ(θ + τ ) 0 0 0 z t (θ)dθ . The function F is re-written in the new variable z t as F (x, ε, t) = (z t (0) 1 z t (0) 2 -η(1 + z t (0) 2 ) 3 z 3 t (-τ ) 1 + I(t), 0) T , i.e. it is a non-linear functional of z t [15, 16] F [z t , t] = 0 -τ 0 -τ ω 2 (θ, θ )z t (θ) ⊗ z t (θ )dθdθ + 0 -τ 0 -τ 0 -τ ω 3 (θ, θ , θ )z t (θ) ⊗ z t (θ ) ⊗ z t (θ )dθdθ dθ +(I(t), 0) t with corresponding tensors ω 2 ∈ R 2×2 , ω 3 ∈ R 2×2×2
and the dyadic product ⊗ . Moreover to gain a compact description the connection function X o (θ) was introduced ad hoc in Eq. ( 4) as

X o (θ) = 0 for -τ ≤ θ < 0, X o (0) = I 2
with the 2 × 2 identity matrix I 2 [START_REF] Hale | Introduction to functional differential equations[END_REF][START_REF] Wischert | [END_REF]. Consequently, considering the functionals L[z t ] and F [z t , t] and the connection function X o (θ) the original scalar problem is reformulated in the infinite-dimensional function z t and the system obeys Eq. ( 4).

The subsequent mathematical treatment is structured as follows. We first solve the linear eigenvalue problem in (4), i.e. F = 0, and project the nonlinear dynamics onto the corresponding (linear) eigenvectors. The augmented system is constructed in such a way that a subset of the eigenspectrum exhibits a vanishing real part and the complementary subset has non-vanishing real parts. Hence it is possible to split the dynamics into two parts, namely the center and the stable part. To reduce the dimension of the still infinite-dimensional system, we introduce a nonautonomous center manifold approach and treat the random input similarly to the nonlinear part. In more detail, the linear eigenspectrum of A is defined by the characteristic equation λ = -1 + exp(-λτ ) with the eigenvalues λ ∈ C. We find λ 0 = 0 plus infinitely many complex conjugate roots λ n , n > 0 with Re(λ n ) < 0.

The corresponding eigenbasis of A is called Φ(θ). Consequently, the phase space splits into a center space U spanned by vectors with the eigenbasis Φ u (θ) corresponding to λ 0 and the stable space S with the corresponding eigenbasis Φ s (θ). The adjunct eigenvalue problem for t → -t yields the adjunct eigenbasis Ψ(θ) that is bi-orthonormal to Φ. To define the metric in the Banach space, a specific bilinear form (•, •) [START_REF] Hale | Introduction to functional differential equations[END_REF][START_REF] Wischert | [END_REF] defines the equivalent of a scalar product in a vector space. We find (Ψ u , Φ u ) = I 2 with the eigenbasis Φ u (θ) ∈ U of the adjunct problem. Accordingly, the center and stable eigenbases are bio-orthogonal with (Ψ u , Φ s ) = 0. Projections onto the center space are given by the projector operator P (•) ≡ Φ u (Ψ u , •) [START_REF] Hale | Introduction to functional differential equations[END_REF][START_REF] Wischert | [END_REF] which reads for ( 1)

P (φ(Θ)) = (1 + τ ) -1 φ(0) + 0 -τ φ(s)ds .
(

) 5
Then one can decompose z t ∈ U × S into the center contribution (u(t), ε) T = P (z t ) ∈ U with the order parameter u ∈ R and a stable contribution s t (θ) = (I -P )(z t ) ∈ S, i.e. z t (θ) = (u(t), ε) T + s t (θ). Consequently the corresponding projected dynamics in U and the complementary space S are given by

du(t) dt = (1 + τ ) -1 F [u + s t , ε, t] , dε(t) dt = 0 (6) d dt s t (θ) = A(s t ) + X o - 1 1 + τ F [u + s t , ε, t] .( 7
)
This formulation ( 6), (7) of the original problem (1) is still infinite dimensional, but allows for a reduction in dimension by center manifold theory, where the dynamics of the stable projection modes are expressed as functions of the center projection modes, i.e. s t (θ, t) = s t (u, ε, θ, t).

In autonomous delayed systems the existence of this relation has been proven mathematically [START_REF] Diekmann | Lecture Notes in Mathematics[END_REF]. Motivated by the existence of stochastic center manifolds in non-delayed systems [19], the present work proposes the following approximation scheme. The small random input is assumed to add a small perturbation to the center manifold obtained for the deterministic (i.e. κ = 0) delayed system, which is independent of the center manifold. Specifically, we assume that the stochastic variation of the center manifold just depends on the delays but is independent of the order parameter u and the control parameter, i.e.

s t (θ) = h det (u, ε, θ) + h t (θ, t) (8)
with the deterministic center manifold contribution h det and the stochastic perturbation h t . In addition, we assume that the stable (nonlinear) projection modes s t fully evolve in the stable (linear) subspace, i.e. P (s t ) = 0. For later computations, it is useful to implement this assumption by P (h det ) = 0 and h t = (I -P )H t with an auxilliary function H t = H(t + θ) ∈ C B . This auxilliary function will be defined later.

We first compute the deterministic center manifold h det according to standard techniques [START_REF] Hale | Introduction to functional differential equations[END_REF][START_REF] Wischert | [END_REF], i.e. s t = h det . It is important to note that, by virtue of the extension of (1) to the augmented system in Eq. ( 3), the relation s t = h det is exact [START_REF] Diekmann | Lecture Notes in Mathematics[END_REF] for all values of the control parameter ε in the absence of noise, i.e. κ = 0. Inserting Eqs. ( 6), ( 7) into (8) yields the partial differential equation for h det in u and θ

1 1 + τ ∂h det ∂u F [u + h det ] = A(h det ) + X o - 1 1 + τ F [u + h det ] .
To solve this equation, typically one considers the activity being close to the resting state u = 0 and chooses the bi-polynomial ansatz

h det = h 2 (u 2 , uε, ε 2 , θ) + h 3 (u 3 , u 2 ε, uε 2 , ε 3 , θ) + • • • .
Assuming that the order parameter has the same order of magnitude as the control parameter, we introduce the scaling u ∼ O(ε) which defines the order of the terms to h n ∼ O(ε n ). Consequently, for small order parameters u it is sufficient to compute just a few lower order terms. Applying the aforementioned necessary condition P (h n ) = 0 up to order O(ε 4 ), the deterministic center manifold reads

h det ≈ (aε + b 3 ε 2 + b 4 ε 3 )u + (d 3 + d 4 ε)u 3 with functions a = a(θ, τ ), b i = b i (θ, τ ), d i = d i (θ, τ
). In addition a, b i , d i → 0 for τ → 0, i.e. the center manifold vanishes for vanishing delay. This reflects the fact that there is no time-scale separation in the dynamics of the non-delayed system. Now let us consider the non-autonmous center manifold perturbation h t . Assuming that the stochastic perturbations are small compared to u and ε, i.e. h t , κ ∼ O(ε 2 ), and inserting Eqs. ( 6), ( 7) into (8) utilizing ds t /dt = (∂h det /∂u)(du/dt) + ∂h t /∂t, the dynamics of the stochastic center manifold contribution h t separates from the autonomous dynamics at lowest order O(ε 2 ) and obeys

∂h t ∂t = A(h t ) + κ X o - 1 1 + τ ξ(t) . (9)
Then recalling the ansatz h t = (I -P )H(t + θ) yields the linear delay equation

dH(t + θ)/dt = L[H] + κξ(t) . ( 10
)
with L[H] = -H(t) + H(t -τ ). It is interesting to mention that (10) is valid for all delayed systems exhibiting a one-dimensional normal form (proof not shown). Hence, the stochastic perturbation of the center manifold h t is independent of the nonlinear interactions in the system. The solution of Eq. ( 10) reads

H(t) = κ t 0 H 0 (t -s)dW (s) , t ≥ 0 , H 0 (t) = 1 2π(1 + τ ) + ∞ n=1 exp(λ n t) 1 + τ exp(-λ n τ ) + c.c. ( 11
)
for initial conditions H(θ) = 0, -τ ≤ θ ≤ 0. Here W (t) is the Wiener process corresponding to the random input ξ(t) and H 0 (t) is the characteristic function of (10) and λ n , Re(λ n ) < 0 are the characteristic roots of the linear problem. Essentially, inserting H(t) into h t = (I -P )H(t + τ ) utilizing the definition of the projector P in Eq. ( 5) gives explicitely the non-autonomous perturbation of the center manifold

h t (θ, t) = κ t+θ 0 H 0 (t + θ -s)dW (s) - κ 1 + τ t 0 H 0 (t -s)dW (s) - κ 1 + τ 0 -τ t+s 0 H 0 (t + s -r)dW (r)ds. ( 12
)
With s t = h det + h t Eq. ( 6) becomes the final order parameter equation

du(t) dt = κ 1 + τ ξ(t) + A 0 Z(t) + B 0 Z 3 (t) +(A 1 + B 1 Z(t) + CZ 2 (t))u + (A 3 + B 3 Z 2 (t))u 3 +A 5 u 5 + A 7 u 7 + A 9 u 9 , (13)
where

Z(t) = h t (t, -τ ) and A n (ε, τ ), B n (ε, τ ), C(ε, τ ) are constants.
We observe that vanishing noise κ = 0 sets Z(t) = 0 and the stable dynamics obeys the deterministic center manifold with h = h det . Additional numerical simulations of the original and the reconstructed solution in the absence of noise have shown a perfect agreement of x(t) and u(t) (result not shown). This finding, that the deterministic center manifold does not affect the dynamics of the original system, is in accordance with a previous study [START_REF] Wischert | [END_REF].

According to (11) and ( 12), the center manifold perturbation Z evolves on infinite many time scales and represents a coloured-noise process. Equation ( 13) reveals that the nonlinear delayed system responds to the additive random process ξ(t) by generating coloured noise Z(t) that modifies the original additive input and renders the system parameters stochastic, i.e. introduces multiplicative noise. This multiplicative noise is well-known to produce noiseinduced transitions and we understand immediately the shift observed in Fig. 1. For clarification, the multiplicative noise originates from the stochasticity of the stable modes s t which couples into the order parameter dynamics. We point out that this novel mechanism is generic in the sense that it occurs in all delayed systems subjected to additive noise which are close to an non-oscillatory instability. Moreover, the specific effect of additive noise on the system depends strongly on the nonlinear order of the system. For instance, in delayed systems involving a quadratic nonlinearity on the center manifold the additive noise induces an additional constant [20] and the center manifold in the absence of noise affects the deterministic dynamics as shown in a previous study [START_REF] Schanz | [END_REF].

To evaluate the proposed reduction scheme, recall the split of z t (θ) into a center manifold and a stable manifold contribution. Applying the center manifold reduction scheme, for θ = 0 this split reads z t (θ = 0) = u(t) + h det (u, ε, 0) + h t (t, 0) = x(t) where the last equation results from the general definition of z t . This relation gives the exact rule of reconstruction of the original process x(t) by u(t) and the corresponding center manifold. Now a reduced description x(t) ≈ u(t), where u obeys Eq. ( 13), is valid if h(t, 0) = h det (u, ε, 0)+h t (t, 0) u(t), i.e. the nondelayed order parameter equation ( 13) approximates well the original systems dynamics of Eq. ( 1). The variance of the stable manifold h 2 (t, 0) , i.e. the reconstruction error made by the reduction of x to u, reads

h 2 (t, 0) = h 2 det (0) + aκ 2 τ 3 /(1 + τ ) 4 + κ 2 R(τ, 1/λ n )
with a pre-factor a and the remaining term R(τ, 1/λ n ) → 0 for τ → 0. Here • denotes the ensemble average. Since h 2 det (0) → 0 for τ → 0, the reconstruction of x is perfect for vanishing delay or noise. For small but non-vanishing delay or noise, the original delayed dynamics is still wellapproximated by the order parameter equation (13).

Similar to previous studies on the deterministic Hopf bifurcation [START_REF] Wischert | [END_REF][START_REF] Schanz | [END_REF], Fig. 2 compares the original and reconstructed solution of Eq. ( 1) and ( 13) by plotting a simulated single path of x(t) and u(t). It reveals a very good fit of x(t) and u(t) which affirms the validity of the proposed reduction scheme.

To gain even deeper insight into the reduced dynamics (13), Fig. 3 compares the numerically estimated stationary probability density functions (PDF) of the original delayed process (1) and the non-delayed order parameter process (13) for various noise strengths κ and delays τ . The corresponding PDFs show very good accordance for τ = 0.5 (Fig. 3(a)), whereas the larger delay τ = 1.0 shown in Fig. 3(b) yields a larger but still reasonable reconstruction error. This result confirms quantitatively the validity of the employed center manifold reduction scheme for small delays. Moreover, we observe that the magnitude of the maxima of the PDFs decrease while increasing the noise strength κ, cf. dashed line in Fig. 3(a). Figure 4 explores this shift in more detail by plotting the location of maxima of the PDFs for various control parameters and reveals a noiseinduced shift to larger values of ε (solid lines and symbols). This finding affirms further the stabilization effect of the additive random process shown in Fig. 1 in the presence of delays.

To quantify the shift of the maxima analytically, we introduce a new random replacement process Z 0 (t) ≈ Z(t) that shares the first two statistical moments with Z(t) for t → ∞, i.e. Z 0 (t) = 0, Z 2 0 (t) = σ 2 with the variance σ 2 = h 2 t (t, -τ ) . This variance is small for small delays and noise strengths (proof not shown). The advantage of introducing this approximative process is the fact that Z(t) is rather complex involving many time scales and Z 0 allows for simpler analytical treatment compared to the original process. We assume that Z 0 (t) obeys the Ornstein-Uhlenbeck process dZ 0 = -(Z 0 /σ 2 )dt + dW 0 (t) with the Wiener process W 0 (t), dW 0 (t) = 0, dW 0 (t)dW 0 (t ) = 2δ(t -t )dtdt . By virtue of the small values of σ 2 , we explore the time scale separation between u and Z 0 and apply an adiabatic elimination scheme [21] to Eq. ( 13). This allows to derive the Fokker-Planck equation for the

with A 1,shif t = Cσ 2 , A 3,shif t = B 3 σ 2 , C and B 3 taken from Eq. ( 13) and D = κ 2 /(1 + τ ) 2 . We observe a noise-dependent additive constant in the linear and cubic terms, and hence a shift of the stationary state of the system by A 1,shif t . In addition C = C(ε, τ ) → 0 for τ → 0, i.e. the noise-induced effect vanishes for τ = 0. Since C < 0, it is A 1,shif t < 0 and the effective stability threshold is shifted to larger values of u. It is easy to find analytically the stationary solution P s (u) of ( 14) and Fig. 4(dashed lines) shows the locations x n of the local maxima of P s (u). As in the numerical results, the delay induces a shift of the probability density to larger magnitudes. Figure 4 also reveals that the delay-induced shift of the original system (symbols) is larger than the shifts found analytically (lines) for larger ε, i.e. the analytical results underestimate the delay-induced shift.

In summary, the work discovers a delay-induced shift of stability in delayed systems driven by additive uncorrelated noise. To explain this phenomenon, we introduce an analytical approach to derive a stochastic non-delayed order parameter equation for a scalar SDDE. This new equation involves both white and coloured noises. For the delayed pitchfork system, a comparison of probability densities reveal close numerical agreement of the nondelayed order parameter equation and the original system. A brief analytical discussion of the probability density of the non-delayed order parameter dynamics affirms the results. Importantly both delay and additive noise affects the presence of a stationary solution at the origin close to

Fig. 2 :

 2 Fig. 2: Comparison of original and reconstructed stochastic paths. The original trajecory x(t) (red) has been computed by numerical integration of Eq. (1) and the reconstructed trajectory u(t) (black) obeys Eq. (13). The inset shows the fulllength trajectories while the large panel presents a focus on a small time interval. Parameters are = 0.2, τ = 0.5, κ = 0.02 and dt = 0.0125, other parameters are taken from Fig. 1.

Fig. 3 :

 3 Fig.3: The order parameter u reconstructs well the original amplitude x in the stationary state near the bifurcation (ε = 0.3). The PDFs Ps(u) and Ps(x) are computed from numerical solutions of Eq. (13) (solid) and (1) (dotted), respectively, for (a) τ = 0.5, (b) τ = 1.0. Noise strengths are κ = 0.005 (green), κ = 0.01 (red) and κ = 0.015 (black). In (a) the PDF maxima are linked (dashed line) to show the shift to lower amplitudes as κ increases. The numerical scheme is identical to the scheme used in Fig.1.

Fig. 4 :

 4 Fig. 4: The location of the maxima of the stationary probability Ps(u) is shifted to larger values in the presence of delay. The locations are estimated numerically from the probility density gained by numerical simulations of the original delay equation (1) (circles and diamonds) and the derived non-delayed equation (13) (solid lines). For comparison, the dashed lines denote the analytical maxima xn gained from Eq. (14).

  probability density P (u, t) of the order parameter u(t) to + A 1,shif t )u + (A 3 + A 3,shif t )u 3 +A 5 u 5 + A 7 u 7 + A 9 u 9 P (u, t) + D ∂ 2 ∂u 2 P (u, t)

	∂P (u, t) ∂t	= -	∂ ∂u	(A 1

JL and AL would like to thank NSERC and FQRNT for support. AH acknowledges funding from the the European Research Council for support under the European Union's Seventh Framework Programme (FP7/2007-2013)/ ERC grant agreement no. 257253.