Résumé

Random polytopes have constituted some of the central objects of stochastic geometry for more than 150 years. They are in general generated as convex hulls of a random set of points in the Euclidean space. The study of such models requires the use of ingredients coming from both convex geometry and probability theory. In the last decades, the study has been focused on their asymptotic properties and in particular expectation and variance estimates. In several joint works with Tomasz Schreiber and J. E. Yukich, we have investigated the scaling limit of several models (uniform model in the unit-ball, uniform model in a smooth convex body, Gaussian model) and have deduced from it limiting variances for several geometric characteristics including the number of k-dimensional faces and the volume. In this paper, we survey the most recent advances on these questions and we emphasize the particular cases of random polytopes in the unit-ball and Gaussian polytopes.

Asymptotic properties of random polytopes

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            <title>Asymptotic properties of random polytopes</title></titles>
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Random polytopes have constituted some of the central objects of stochastic geometry for more than 150 years. They are in general generated as convex hulls of a random set of points in the Euclidean space. The study of such models requires the use of ingredients coming from both convex geometry and probability theory. In the last decades, the study has been focused on their asymptotic properties and in particular expectation and variance estimates. In several joint works with Tomasz Schreiber and J. E. Yukich, we have investigated the scaling limit of several models (uniform model in the unit-ball, uniform model in a smooth convex body, Gaussian model) and have deduced from it limiting variances for several geometric characteristics including the number of k-dimensional faces and the volume. In this paper, we survey the most recent advances on these questions and we emphasize the particular cases of random polytopes in the unit-ball and Gaussian polytopes.

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Asymptotic properties of random polytopes Pierre Calka 2nd conference on Geometric Science of Information ´Ecole Polytechnique, Paris-Saclay, 28 October 2015 default Outline Random polytopes: an overview Main results: variance asymptotics Sketch of proof: Gaussian case Joint work with Joseph Yukich (Lehigh University, USA) & Tomasz Schreiber (Toru´n University, Poland) default Outline Random polytopes: an overview Uniform polytopes Gaussian polytopes Expectation asymptotics Main results: variance asymptotics Sketch of proof: Gaussian case default Uniform polytopes Binomial model K := convex body of Rd (Xk,k ∈ N∗):= independent and uniformly distributed in K Kn := Conv(X1, · · · , Xn), n ≥ 1 K50, K ball K50, K square default Uniform polytopes Binomial model K := convex body of Rd (Xk,k ∈ N∗):= independent and uniformly distributed in K Kn := Conv(X1, · · · , Xn), n ≥ 1 K100, K ball K100, K square default Uniform polytopes Binomial model K := convex body of Rd (Xk,k ∈ N∗):= independent and uniformly distributed in K Kn := Conv(X1, · · · , Xn), n ≥ 1 K500, K ball K500, K square default Uniform polytopes Poissonian model K := convex body of Rd Pλ, λ > 0:= Poisson point process of intensity measure λdx Kλ := Conv(Pλ ∩ K) K500, K ball K500, K square default Gaussian polytopes Binomial model Φd (x) := 1 (2π)d/2 e− x 2/2, x ∈ Rd, d ≥ 2 (Xk, k ∈ N∗):= independent and with density Φd Kn := Conv(X1, · · · , Xn) Poissonian model Pλ, λ > 0:= Poisson point process of intensity measure λΦd(x)dx Kλ := Conv(Pλ) default Gaussian polytopes K50 K100 K500 default Gaussian polytopes: spherical shape K50 K100 K500 default Asymptotic spherical shape of the Gaussian polytope Geffroy (1961) : dH(Kn, B(0, 2 log(n))) → n→∞ 0 a.s. K50000 default Expectation asymptotics Considered functionals fk(·) := number of k-dimensional faces, 0 ≤ k ≤ d Vol(·) := volume B. Efron’s relation (1965): Ef0(Kn) = n 1 − EVol(Kn−1) Vol(K) Uniform polytope, K smooth E[fk(Kλ)] ∼ λ→∞ cd,k ∂K κ 1 d+1 s ds λ d−1 d+1 κs := Gaussian curvature of ∂K Uniform polytope, K polytope E[fk(Kλ)] ∼ λ→∞ c′ d,kF(K) logd−1 (λ) F(K) := number of flags of K Gaussian polytope E[fk(Kλ)] ∼ λ→∞ c′′ d,k log d−1 2 (λ) A. R´enyi & R. Sulanke (1963), H. Raynaud (1970), R. Schneider & J. Wieacker (1978), F. Affentranger & R. Schneider (1992) default Outline Random polytopes: an overview Main results: variance asymptotics Uniform model, K smooth Uniform model, K polytope Gaussian model Sketch of proof: Gaussian case default Uniform model, K smooth K := convex body of Rd with volume 1 and with a C3 boundary κ := Gaussian curvature of ∂K lim λ→∞ λ−(d−1)/(d+1) Var[fk(Kλ)] = ck,d ∂K κ(z)1/(d+1) dz lim λ→∞ λ(d+3)/(d+1) Var [Vol(Kλ)] = c′ d ∂K κ(z)1/(d+1) dz (ck,d , c′ d explicit positive constants) M. Reitzner (2005): Var[fk (Kλ)] = Θ(λ(d−1)/(d+1) ) default Uniform model, K polytope K := simple polytope of Rd with volume 1 i.e. each vertex of K is included in exactly d facets. lim λ→∞ log−(d−1) (λ)Var[fk(Kλ)] = cd,kf0(K) lim λ→∞ λ2 log−(d−1) (λ)Var[Vol(Kλ)] = c′ d,k f0(K) (ck,d , c′ k,d explicit positive constants) I. B´ar´any & M. Reitzner (2010): Var[fk (Kλ)] = Θ(log(d−1) (λ)) default Gaussian model lim λ→∞ log− d−1 2 (λ)Var[fk(Kλ)] = ck,d lim λ→∞ log−k+ d+3 2 (λ)Var[Vol(Kλ)] = c′ k,d E Vol(Kλ) Vol(B(0, 2 log(n))) = λ→∞ 1 − d log(log(λ)) 4 log(λ) + O 1 log(λ) (ck,d , c′ k,d explicit positive constants) D. Hug & M. Reitzner (2005), I. B´ar´any & V. Vu (2007): Var[fk (Kλ)] = Θ(log(d−1)/2 (λ)) default Outline Random polytopes: an overview Main results: variance asymptotics Sketch of proof: Gaussian case Calculation of the expectation of fk(Kλ) Calculation of the variance of fk(Kλ) Scaling transform default Calculation of the expectation of fk(Kλ) 1. Decomposition: E[fk(Kλ)] = E   x∈Pλ ξ(x, Pλ)   ξ(x, Pλ) := 1 k+1 #k-face containing x if x extreme 0 if not 2. Mecke-Slivnyak formula E[fk(Kλ)] = λ E[ξ(x, Pλ ∪ {x})]Φd (x)dx 3. Limit of the expectation of one score default Calculation of the variance of fk(Kλ) Var[fk (Kλ)] = E   x∈Pλ ξ2 (x, Pλ) + x=y∈Pλ ξ(x, Pλ)ξ(y, Pλ)   − (E[fk (Kλ)]) 2 = λ E[ξ2 (x, Pλ ∪ {x})]Φd(x)dx + λ2 E[ξ(x, Pλ ∪ {x, y})ξ(y, Pλ ∪ {x, y})]Φd (x)Φd (y)dxdy − λ2 E[ξ(x, Pλ ∪ {x})]E[ξ(y, Pλ ∪ {y})]Φd (x)Φd (y)dxdy = λ E[ξ2 (x, Pλ ∪ {x})]Φd(x)dx + λ2 ”Cov”(ξ(x, Pλ ∪ {x}), ξ(y, Pλ ∪ {y}))Φd (x)Φd (y)dxdy default Scaling transform Question : Limits of E[ξ(x, Pλ)] and ”Cov”(ξ(x, Pλ), ξ(y, Pλ)) ? Answer : definition of limit scores in a new space ◮ Critical radius Rλ := 2 log λ − log(2 · (2π)d · log λ) ◮ Scaling transform : Tλ : Rd \ {0} −→ Rd−1 × R x −→ Rλ exp−1 d−1 x |x|, R2 λ(1 − |x| Rλ ) expd−1 : Rd−1 ≃ Tu0 Sd−1 → Sd−1 exponential map at u0 ∈ Sd−1 ◮ Image of a score : ξ(λ)(Tλ(x), Tλ(Pλ)) := ξ(x, Pλ) ◮ Convergence of Pλ : Tλ(Pλ) D → P o`u P : Poisson point process in Rd−1 × R of intensity measure ehdvdh default Action of the scaling transform Π↑ := {(v, h) ∈ Rd−1 × R : h ≥ v 2 2 } Π↓ := {(v, h) ∈ Rd−1 × R : h ≤ − v 2 2 } Half-space Translate of Π↓ Sphere containing O Translate of ∂Π↑ Convexity Parabolic convexity Extreme point (x + Π↑) not fully covered k-face of Kλ Parabolic k-face RλVol Vol default Limiting picture Ψ := x∈P(x + Π↑) In red : image of the balls of diameter [0, x] where x is extreme default Limiting picture Φ := x∈Rd−1×R:x+Π↓∩P=∅(x + Π↓) In green : image of the boundary of the convex hull Kλ default Thank you for your attention!