On the maximal perimeter of isotropic log-concave probability measures

Abstract: We study the maximal perimeter constant of isotropic log-concave probability measures on $\mathbb{R}<sup>n$. 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 $$Γ_n := \sup{Γ(μ) : μ\text{ is an isotropic log-concave probability measure on } \mathbb{R}<sup>n}.$$ We prove that $Γ_n \leqslant Cn<sup>{3/2}$, where $C&gt;0$ is an absolute constant. This result improves the previously known $O(n<sup>2)$ upper bound. Under additional structural assumptions, we obtain sharp linear bounds of order $O(n)$.