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\le p\le q$${\left(E\right|〈X,e_n〉{|}^p)}^{\frac{1}{p}}\le {\left(E\right|〈X,e_n〉{|}^q)}^{\frac{1}{q}}\le C\frac{q}{p}{\left(E\right|〈X,e_n〉{|}^p)}^{\frac{1}{p}},$ whenever X is a random vector uniformly distributed on any convex body $K\subseteq R^n$ and ${\left(e_i\right)}_{i=1}^n$ 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 $K\cap Z^n$ for any convex body $K\subseteq R^n$ 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 $Ew\left(K_N\right)\sim w\left(Z_{\log N}\right(K\left)\right)$ for any convex body K containing the origin in its interior, where $K_N$ is the centrally symmetric random polytope $K_N=\text{conv}\{\pm X_1,\dots ,\pm X_N\}$ generated by independent random vectors uniformly distributed on K, $Z_p\left(K\right)$ is the $L_p$-centroid body of K for any $p\ge 1$, and $w(\cdot )$ denotes the mean width.