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

Borell's inequality states the existence of a positive absolute constant $C>0$ such that for every 1≤ p≤ q $$ ( E| X, e_n|^p)^1/p≤( E| X, e_n|^q)^1/q≤ Cq/p( E| X, e_n|^p)^1/p, $$ whenever $X$ is a random vector uniformly distributed in any convex body $K⊆ R^n$ containing the origin in its interior and $(e_i)ᵢ₌₁^n$ is the standard canonical basis in $ 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 $K∩ Z^n$ for any convex body $K⊆ R^n$ 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 $ E w(K_N)~ w(Zlog N(K))$ for any convex body $K$ containing the origin in its interior, where $K_N$ is the centrally symmetric random polytope $K_N=conv\{± X_1,…,± X_N\}$ generated by independent random vectors uniformly distributed on $K$ and $w(·)$ denotes the mean width.