Poincar, inequality in mean value for Gaussian polytopes

• 成果类型: Article
• 署名作者: Fleury, B.
• 署名单位: Sorbonne Universite; Universite Paris Cite
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-010-0318-3
• 发表日期: 2012
• 页码: 141-178
• 关键词: CENTRAL-LIMIT-THEOREM; convex

摘要

Let K-N = [+/- G1, ...., +/- G(N)] be the absolute convex hull of N independent standardGaussian random points in R-n with N >= n. We prove that, for any 1- Lipschitz function f : R-n -> R, the polytope K-N satisfies the following Poincare inequality in mean value: E-omega integral(KN(omega))(f (x) - 1/vol(n) (K-N (omega))integral(KN(omega)) f (y)dy)(2) dx <= C/n E-omega integral(KN(omega)) |x|(2)dx where C > 0 is an absolute constant. This Poincare inequality is the one suggested by a conjecture of Kannan, Lovasz and Simonovits for general convex bodies. Moreover, we prove in mean value that the volume of the polytope K-N is concentrated in a subexponential way within a thin Euclidean shell with the optimal dependence of the dimension n. An important tool of the proofs is a representation of the law of (G(1), ..., G(n)) conditioned by the event that the convex hull of G(1), ..., G(n) is a (n - 1)- face of KN. As an application, we also get an estimate of the number of (n - 1)- faces of the polytope K-N, valid for every N >= n.