On the maximal perimeter of isotropic log-concave probability measures

TL;DR

This paper investigates the maximum perimeter constant of isotropic log-concave measures in high-dimensional spaces, establishing an improved upper bound of order $n^{3/2}$ and exploring conditions for linear bounds.

Contribution

The paper proves a new upper bound of $Cn^{3/2}$ for the maximal perimeter constant of isotropic log-concave measures, improving previous $O(n^2)$ bounds.

Findings

Established an upper bound of $Cn^{3/2}$ for $ ext{Gamma}_n$

Improved from previous $O(n^2)$ bounds

Under structural assumptions, obtained linear $O(n)$ bounds

Abstract

We study the maximal perimeter constant of isotropic log-concave probability measures on $\mathbb{R}^n$. For a measure $\mu$, this quantity, denoted by $\Gamma(\mu)$, is defined as the supremum of the $\mu$-perimeter over all convex bodies and measures the largest possible boundary contribution of convex sets with respect to $\mu$. Let $$\Gamma_n := \sup\{\Gamma(\mu) : \mu \text{ is an isotropic log-concave probability measure on } \mathbb{R}^n\}.$$ We prove that $\Gamma_n \leqslant Cn^{3/2}$, where $C>0$ is an absolute constant. This result improves the previously known $O(n^2)$ upper bound. Under additional structural assumptions, we obtain sharp linear bounds of order $O(n)$.