Horocyclic Brunn-Minkowski inequality
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866913217624670208 |
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| author | Assouline, Rotem Klartag, Bo'az |
| author_facet | Assouline, Rotem Klartag, Bo'az |
| contents | Given two non-empty subsets $A$ and $B$ of the hyperbolic plane $\mathbb{H}^2$, we define their horocyclic Minkowski sum with parameter $λ=1/2$ 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 $0 < λ<1$ is defined analogously. We prove that when $A$ and $B$ are Borel-measurable, $$ \sqrt{ Area( [A:B]_λ )} \geq (1-λ) \cdot \sqrt{ Area(A) } + λ\cdot \sqrt{ Area(B) }, $$ 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 Prékopa-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_09826 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Horocyclic Brunn-Minkowski inequality Assouline, Rotem Klartag, Bo'az Metric Geometry Differential Geometry Given two non-empty subsets $A$ and $B$ of the hyperbolic plane $\mathbb{H}^2$, we define their horocyclic Minkowski sum with parameter $λ=1/2$ 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 $0 < λ<1$ is defined analogously. We prove that when $A$ and $B$ are Borel-measurable, $$ \sqrt{ Area( [A:B]_λ )} \geq (1-λ) \cdot \sqrt{ Area(A) } + λ\cdot \sqrt{ Area(B) }, $$ 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 Prékopa-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. |
| title | Horocyclic Brunn-Minkowski inequality |
| topic | Metric Geometry Differential Geometry |
| url | https://arxiv.org/abs/2208.09826 |