Borell's inequality and mean width of random polytopes via discrete inequalities

TL;DR

This paper extends Borell's inequality to discrete settings involving lattice points in convex bodies and applies it to analyze the mean width of random polytopes, connecting geometric and probabilistic properties.

Contribution

It introduces a discrete version of Borell's inequality for lattice points in convex bodies and uses it to derive estimates for the mean width of random polytopes.

Findings

Discrete Borell's inequality for lattice points

Estimate of mean width of random polytopes

Connection between discrete inequalities and geometric properties

Abstract

Borell's inequality states the existence of a positive absolute constant $C>0$ such that for every $1\leq p\leq q$ $$ \left(\mathbb E|\langle X, e_n\rangle|^p\right)^\frac{1}{p}\leq\left(\mathbb E|\langle X, e_n\rangle|^q\right)^\frac{1}{q}\leq C\frac{q}{p}\left(\mathbb E|\langle X, e_n\rangle|^p\right)^\frac{1}{p}, $$ whenever $X$ is a random vector uniformly distributed on any convex body $K\subseteq\mathbb R^n$ and $(e_i)_{i=1}^n$ is the standard canonical basis in $\mathbb R^n$. In this paper, we will prove a discrete version of this inequality, which will hold whenever $X$ is a random vector uniformly distributed on $K\cap\mathbb Z^n$ for any convex body $K\subseteq\mathbb R^n$ containing the origin in its interior. We will also make use of such discrete version to obtain discrete inequalities from which we can recover the estimate $\mathbb E w(K_N)\sim w(Z_{\log N}(K))$ for any convex body $K$ containing the origin in its interior, where $K_N$ is the centrally symmetric random polytope $K_N=\textrm{conv}\{\pm X_1,\ldots,\pm X_N\}$ generated by independent random vectors uniformly distributed on $K$, $Z_{p}(K)$ is the $L_p$-centroid body of $K$ for any $p\geq1$, and $w(\cdot)$ denotes the mean width.