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

Abstract

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 $μ$ 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 $λ\in(0,1)$, $$μ(λK+(1-λ)L)^{c_n} \geq λμ(K)^{c_n}+(1-λ)μ(L)^{c_n},$$ where $c_n\geq c/n^3\ln n$ for some absolute constant $c>0$. Secondly, we study the maximal perimeter $Γ(μ)$ of an isotropic log-concave measure $μ$, without symmetry assumptions. We prove that $$Γ_n = \sup\{Γ(μ): \ μ\ \mbox{is an isotropic log-concave measure on } \mathbb{R}^n \} \approx n.$$ A key ingredient in both our proofs is a bound due to Eldan and Klartag (2008), which states that $$\int_{\mathbb{R}^n} |\nablaψ|\,dμ\leq Cn$$ for every isotropic log-concave probability measure $μ$ on $\mathbb{R}^n$ with density $e^{-ψ}$. 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.