Poincaré inequality in mean value for Gaussian polytopes

Abstract

Let K N = [±G 1, . . . , ±G N ] be the absolute convex hull of N independent standard Gaussian random points in \({\mathbb R^n}\) with N ≥ n. We prove that, for any 1-Lipschitz function \({f:\mathbb R^n\rightarrow\mathbb R}\), the polytope K N satisfies the following Poincaré inequality in mean value: $$\mathbb {E}_{\omega} \int\limits_{K_N(\omega)} \left( f(x) - \frac{1}{\textup{vol}_n\left(K_N(\omega)\right)} \int\limits_{K_n(\omega)}f(y)dy \right)^2 dx \leq \frac{C}{n} \mathbb E_{\omega} \int\limits_{K_N(\omega)}|x|^2dx$$ where C > 0 is an absolute constant. This Poincaré inequality is the one suggested by a conjecture of Kannan, Lovász 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 K N ”. 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.