Pré-Publication, Document De Travail Année : 2026

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

Résumé

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. Our main result asserts 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 $c>0$. A key ingredient in our proof is the bound $$\int_{\mathbb{R}^n} |\nabla\psi|\,dμ\leq Cn$$ that we establish for isotropic log-concave probability measures $μ$ 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.

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hal-05614148 , version 1 (06-05-2026)

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Alexandros Eskenazis, Apostolos Giannopoulos, Natalia Tziotziou. Functional perimeter and the dimensional Brunn-Minkowski inequality for log-concave measures. 2026. ⟨hal-05614148⟩
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