Functional perimeter and the dimensional Brunn-Minkowski inequality for log-concave measures
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917459016024064 |
|---|---|
| author | Eskenazis, Alexandros Giannopoulos, Apostolos Tziotziou, Natalia |
| author_facet | Eskenazis, Alexandros Giannopoulos, Apostolos Tziotziou, Natalia |
| contents | 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 $$\int_{\mathbb{R}^n} |\nablaψ|\,dμ\leq Cn$$ that we establish for isotropic log-concave probability measures $μ$ on $\mathbb{R}^n$ with density $e^{-ψ}$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02747 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Functional perimeter and the dimensional Brunn-Minkowski inequality for log-concave measures Eskenazis, Alexandros Giannopoulos, Apostolos Tziotziou, Natalia Metric Geometry Functional Analysis 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 $$\int_{\mathbb{R}^n} |\nablaψ|\,dμ\leq Cn$$ that we establish for isotropic log-concave probability measures $μ$ on $\mathbb{R}^n$ with density $e^{-ψ}$, 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. |
| title | Functional perimeter and the dimensional Brunn-Minkowski inequality for log-concave measures |
| topic | Metric Geometry Functional Analysis |
| url | https://arxiv.org/abs/2605.02747 |