On the maximal perimeter of isotropic log-concave probability measures

We study the maximal perimeter constant of isotropic log-concave probability measures on ${\mathbb{R}}^n$. For a measure $\mu$, this quantity, denoted by $\mathrm{\Gamma }(\mu )$, 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 $${\mathrm{\Gamma }}_n:=\sup \{\mathrm{\Gamma }(\mu ):\mu \text{ is an isotropic log-concave probability measure on }{\mathbb{R}}^n\}\mathrm{.}$$ We prove that ${\mathrm{\Gamma }}_n⩽C{n}^{3\mathrm{/}2}$, 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 $O(n)$.