Horocyclic Brunn-Minkowski inequality
| Autor: | Assouline, Rotem, Klartag, Bo'az |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Advances in Mathematics, Volume 436, 2024 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.aim.2023.109381 |
| Popis: | Given two non-empty subsets $A$ and $B$ of the hyperbolic plane $\mathbb{H}^2$, we define their horocyclic Minkowski sum with parameter $\lambda=1/2$ as the set $[A:B]_{1/2} \subseteq \mathbb{H}^2$ of all midpoints of horocycle curves connecting a point in $A$ with a point in $B$. These horocycle curves are parameterized by hyperbolic arclength, and the horocyclic Minkowski sum with parameter $0 < \lambda <1$ is defined analogously. We prove that when $A$ and $B$ are Borel-measurable, $$ \sqrt{ Area( [A:B]_{\lambda} )} \geq (1-\lambda) \cdot \sqrt{ Area(A) } + \lambda \cdot \sqrt{ Area(B) }, $$ where $Area$ stands for hyperbolic area, with equality when $A$ and $B$ are concentric discs in the hyperbolic plane. We also prove horocyclic versions of the Pr\'ekopa-Leindler and Borell-Brascamp-Lieb inequalities. These inequalities slightly deviate from the metric measure space paradigm on curvature and Brunn-Minkowski type inequalities, where the structure of a metric space is imposed on the manifold, and the relevant curves are necessarily geodesics parameterized by arclength. Comment: 36 pages, 2 figures. v5: minor changes. Final version, to appear in Advances in Mathematics. v4: The paper has undergone considerable reorganization. The order of the sections has changed, and the focus is now mostly on horocycles without reference to more general path spaces |
| Databáze: | arXiv |
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