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

Abstract: We study the dimensional Brunn-Minkowski inequality for even log-concave probability measures $μ$ on $\mathbb{R}<sup>n$ 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 $\mathbb{R}<sup>n$ and every $λ\in(0,1)$, $$μ(λK+(1-λ)L)^{c_n} \geq λμ(K)<sup>{c_n}+(1-λ)μ(L)<sup>{c_n},$$ where $c_n\geq c/n^3\ln n$ for some absolute constant $c&gt;0$. A key ingredient in our proof is the bound $\int_{\mathbb{R}^n} |\nablaψ|\,dμ\leq Cn$ that we establish for isotropic log-concave probability measures $μ$ on $\mathbb{R}<sup>n$ with density $e<sup>{-ψ}$, which is optimal in terms of the dimension. This estimate yields structural information on the size of sub-level sets of the gradient of $ψ$ 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.