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,en〉|p)1p≤(E|〈X,en〉|q)1q≤Cqp(E|〈X,en〉|p)1p, whenever X is a random vector uniformly distributed on any convex body K⊆Rn and (ei)i=1n is the standard canonical basis in Rn. 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∩Zn for any convex body K⊆Rn 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 Ew(KN)∼w(Zlog⁡N(K)) for any convex body K containing the origin in its interior, where KN is the centrally symmetric random polytope KN=conv{±X1,…,±XN} generated by independent random vectors uniformly distributed on K, Zp(K) is the Lp-centroid body of K for any p≥1, and w(⋅) denotes the mean width.