Volume properties of high-dimensional Orlicz balls

TL;DR

This paper investigates the volume asymptotics of high-dimensional Orlicz balls, explores their spectral gap properties, and analyzes the independence and integrability of linear functionals on these balls.

Contribution

It provides new asymptotic volume estimates for Orlicz balls and verifies a conjecture related to spectral gaps, also studying coordinate independence and linear functional properties.

Findings

Asymptotic volume estimates for Orlicz balls in high dimensions

Verification of a conjecture on spectral gaps for certain Orlicz balls

Analysis of coordinate independence and linear functional integrability

Abstract

We prove asymptotic estimates for the volume of families of Orlicz balls in high dimensions. As an application, we describe a large family of Orlicz balls which verify a famous conjecture of Kannan, Lov{\'a}sz and Simonovits about spectral gaps. We also study the asymptotic independence of coordinates on uniform random vectors on Orlicz balls, as well as integrability properties of their linear functionals.