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

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. Our main result asserts 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 $c>0$. A key ingredient in our proof is the bound $$\int_{\mathbb{R}^n} |\nabla\psi|\,d\mu \leq Cn$$ that we establish for isotropic log-concave probability measures $\mu$ on $\mathbb{R}^n$ with density $e^{-\psi}$, which is optimal in terms of the dimension. This estimate yields structural information on the size of sub-level sets of the gradient of $\psi$ and puts forth a geometric obstruction to further improvements of the Brunn-Minkowski exponent. We also present applications of this estimate to the weighted perimeter of level sets, 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.