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 $\lambda\in(0,1)$, $$\mu(\lambda K+(1-\lambda)L)^{c_n} \geq \lambda \mu(K)^{c_n}+(1-\lambda)\mu(L)^{c_n},$$ where $c_n\geq c/n^3\ln n$ for some absolute constant $c>0$. Secondly, we study the maximal perimeter $\Gamma(\mu)$ of an isotropic log-concave measure $\mu$, without symmetry assumptions. We prove that $$\Gamma_n = \sup\{\Gamma(\mu): \ \mu \ \mbox{is an isotropic log-concave measure on } \mathbb{R}^n \} \approx n.$$ A key ingredient in both our proofs is a bound due to Eldan and Klartag (2008), which states that $$\int_{\mathbb{R}^n} |\nabla\psi|\,d\mu \leq Cn$$ 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.