On the maximal perimeter of isotropic log-concave probability measures
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912873748365312 |
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| author | Brazitikos, Silouanos Giannopoulos, Apostolos Hmadi, Antonios Tziotziou, Natalia |
| author_facet | Brazitikos, Silouanos Giannopoulos, Apostolos Hmadi, Antonios Tziotziou, Natalia |
| contents | We study the maximal perimeter constant of isotropic log-concave probability measures on $\mathbb{R}^n$. For a measure $μ$, this quantity, denoted by $Γ(μ)$, is defined as the supremum of the $μ$-perimeter over all convex bodies and measures the largest possible boundary contribution of convex sets with respect to $μ$. Let $$Γ_n := \sup\{Γ(μ) : μ\text{ is an isotropic log-concave probability measure on } \mathbb{R}^n\}.$$ We prove that $Γ_n \leqslant Cn^{3/2}$, where $C>0$ is an absolute constant. This result improves the previously known $O(n^2)$ upper bound. Under additional structural assumptions, we obtain sharp linear bounds of order $O(n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03831 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the maximal perimeter of isotropic log-concave probability measures Brazitikos, Silouanos Giannopoulos, Apostolos Hmadi, Antonios Tziotziou, Natalia Metric Geometry Functional Analysis Probability Primary 52A38, Secondary 52A40, 60D05, 46B06 We study the maximal perimeter constant of isotropic log-concave probability measures on $\mathbb{R}^n$. For a measure $μ$, this quantity, denoted by $Γ(μ)$, is defined as the supremum of the $μ$-perimeter over all convex bodies and measures the largest possible boundary contribution of convex sets with respect to $μ$. Let $$Γ_n := \sup\{Γ(μ) : μ\text{ is an isotropic log-concave probability measure on } \mathbb{R}^n\}.$$ We prove that $Γ_n \leqslant Cn^{3/2}$, where $C>0$ is an absolute constant. This result improves the previously known $O(n^2)$ upper bound. Under additional structural assumptions, we obtain sharp linear bounds of order $O(n)$. |
| title | On the maximal perimeter of isotropic log-concave probability measures |
| topic | Metric Geometry Functional Analysis Probability Primary 52A38, Secondary 52A40, 60D05, 46B06 |
| url | https://arxiv.org/abs/2602.03831 |