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

Alonso-Gutiérrez, David (Universidad de Zaragoza) ; García-Lirola, Luis C. (Universidad de Zaragoza)

Resumen: 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.

Idioma: Inglés

DOI: 10.1016/j.jco.2025.101993

Año: 2026

Publicado en: JOURNAL OF COMPLEXITY 92 (2026), 101993 [25 pp.]
ISSN: 0885-064X

Financiación: info:eu-repo/grantAgreement/ES/DGA/E48-23R
Financiación: info:eu-repo/grantAgreement/ES/MICIU/PID2021-122126NB-C32
Financiación: info:eu-repo/grantAgreement/ES/MICIU/PID2022-137294NB-I00

Tipo y forma: Article (Published version)

Área (Departamento): Área Análisis Matemático (Dpto. Matemáticas)