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

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.