Maximal surface area of a convex set in $\mathbb{R}^n$ with respect to exponential rotation invariant measures

TL;DR

This paper determines the asymptotic maximal surface area of convex bodies in high-dimensional space under exponential rotation-invariant measures, generalizing known Gaussian measure results.

Contribution

It extends previous bounds for Gaussian measures to a broader class of exponential rotation-invariant measures, providing asymptotic formulas for maximal surface area.

Findings

Maximal surface area scales as $C_p n^{3/4 - 1/p}$ for measure $ u_p$.

Generalizes bounds from Gaussian to exponential rotation-invariant measures.

Provides explicit dependence of constants on parameter $p$.

Abstract

Let $p$ be a positive number. Consider probability measure $\gamma_p$ with density $\varphi_p(y)=c_{n,p}e^{-\frac{|y|^p}{p}}$. We show that the maximal surface area of a convex body in $\mathbb{R}^n$ with respect to $\gamma_p$ is asymptotically equal to $C_p n^{\frac{3}{4}-\frac{1}{p}}$, where constant $C_p$ depends on $p$ only. This is a generalization of Ball's and Nazarov's bounds, which were given for the case of the standard Gaussian measure $\gamma_2$.