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 in any convex body K Rⁿ containing the origin in its interior and (eᵢ) ₈=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 E w (KN) w (Z ₍ (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 and w () denotes the mean width.