On the maximal perimeter of isotropic log-concave probability measures

TL;DR

The paper addresses the growth of the maximal perimeter constant ${Γ_n}$ for isotropic log-concave measures on ${\mathbb{R}^n}$ and proves the key bound ${Γ_n \le C\,n^{3/2}}$, improving the previous quadratic bound. The authors develop a framework based on level sets ${R_t(\mu)}$ and dilation-type inequalities, localizing perimeter estimates to regions where the density is well-controlled, and they leverage Steinhagen's width–inradius relation to achieve the bound. They further establish sharp linear bounds in several structured settings, including uniform measures on isotropic convex bodies, 1-unconditional measures, one-dimensional isotropic distributions, and product measures with bounded marginals, highlighting when linear growth is attainable. These results illuminate how different high-dimensional log-concave measures exhibit distinct perimeter scaling, with implications for convex geometry and high-dimensional probability. The methodology combines geometric analysis of level sets, one-dimensional marginals, and transformations that preserve or regularize density profiles, contributing a nuanced understanding of boundary contributions in high dimensions.

Abstract

We study the maximal perimeter constant of isotropic log-concave probability measures on $\mathbb{R}^n$. For a measure $μ$, this quantity, denoted by $Γ(μ)$, is defined as the supremum of the $μ$-perimeter over all convex bodies and measures the largest possible boundary contribution of convex sets with respect to $μ$. Let $$Γ_n := \sup\{Γ(μ) : μ\text{ is an isotropic log-concave probability measure on } \mathbb{R}^n\}.$$ We prove that $Γ_n \leqslant Cn^{3/2}$, 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 $O(n)$.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Remark 3.3
• Lemma 4.1
• proof
• Lemma 4.2