doi:10.1016/j.jco.2025.101993engAlonso-Gutiérrez, DavidGarcía-Lirola, Luis C.Borell's inequality and mean width of random polytopes via discrete inequalitiesART-2026-145433Borell'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.2026http://zaguan.unizar.es/record/16295910.1016/j.jco.2025.101993http://zaguan.unizar.es/record/162959oai:zaguan.unizar.es:162959info:eu-repo/grantAgreement/ES/DGA/E48-23Rinfo:eu-repo/grantAgreement/ES/MICIU/PID2021-122126NB-C32info:eu-repo/grantAgreement/ES/MICIU/PID2022-137294NB-I00JOURNAL OF COMPLEXITY 92 (2026), 101993 [25 pp.]byhttps://creativecommons.org/licenses/by/4.0/deed.esinfo:eu-repo/semantics/openAccess