The maximum entropy principle and volumetric properties of Orlicz balls

TL;DR

This paper investigates the asymptotic volume of Orlicz balls and their intersections in high dimensions, revealing phase transitions and connecting geometric properties with statistical mechanics principles.

Contribution

It generalizes known results for $ ext{l}_p^d$-balls to Orlicz spaces, introducing a novel approach using maximum entropy principles for volumetric analysis.

Findings

Identifies phase transition in intersection volumes of Orlicz balls in high dimensions

Determines asymptotic volume ratios for 2-concave Orlicz spaces

Connects the geometry of Orlicz balls with statistical mechanics concepts

Abstract

We study the precise asymptotic volume of balls in Orlicz spaces and show that the volume of the intersection of two Orlicz balls undergoes a phase transition when the dimension of the ambient space tends to infinity. This generalizes a result of Schechtman and Schmuckenschl\"ager [GAFA, Lecture notes in Math. 1469 (1991), 174--178] for $\ell_p^d$-balls. As another application, we determine the precise asymptotic volume ratio for $2$-concave Orlicz spaces $\ell_M^d$. Our method rests on ideas from statistical mechanics and large deviations theory, more precisely the maximum entropy or Gibbs principle for non-interacting particles, and presents a natural approach and fresh perspective to such geometric and volumetric questions. In particular, our approach explains how the $p$-generalized Gaussian distribution occurs in problems related to the geometry of $\ell_p^d$-balls, which are Orlicz balls when the Orlicz function is $M(t) = |t|^p$.