Functional perimeter and the dimensional Brunn-Minkowski inequality for log-concave measures

This paper is dedicated to two geometric problems associated to log-concave measures on ${\mathbb{R}}^n$. First, we study the dimensional Brunn-Minkowski inequality for even log-concave probability measures $\mu$ on ${\mathbb{R}}^n$ via an analytic approach based on diffusion operators and gradient estimates. We prove that for every pair of symmetric convex sets $K,L$ in ${\mathbb{R}}^n$ and every $\lambda \in (0,1)$, $$\mu (\lambda K+(1-\lambda )L{)}^{c_n}\ge \lambda \mu (K{)}^{c_n}+(1-\lambda )\mu (L{)}^{c_n},$$ where $c_n\ge c\mathrm{/}{n}^3\ln n$ for some absolute constant $c>0$. Secondly, we study the maximal perimeter $\mathrm{\Gamma }(\mu )$ of an isotropic log-concave measure $\mu$, without symmetry assumptions. We prove that $${\mathrm{\Gamma }}_n=\sup \{\mathrm{\Gamma }(\mu ):\mu isanisotropiclog-concavemeasureon{\mathbb{R}}^n\}\approx n\mathrm{.}$$ A key ingredient in both our proofs is a bound due to Eldan and Klartag (2008), which states that $${\int }_{{\mathbb{R}}^n}\mathrm{\mid }\mathrm{\nabla }\psi \mathrm{\mid }d\mu \le Cn$$ for every isotropic log-concave probability measure $\mu$ on ${\mathbb{R}}^n$ with density $e^{-\psi }$. We also present further applications of this estimate to projections of log-concave functions projections, moment and surface area measures of isotropic log-concave functions, highlighting the central role of the gradient of the logarithmic potential in high-dimensional convexity.