$β$-Uniform Convexity and Divisible Domains
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917818479411200 |
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| author | Pompilio, Amelia |
| author_facet | Pompilio, Amelia |
| contents | Divisible convex sets have long been important in the study of Hilbert geometries. When a divisible convex set is an ellipsoid, the Hilbert geometry it induces is the hyperbolic space. In general, strictly convex divisible domains exhibit negative curvature properties, but only the ellipsoid is a CAT(0) space. The notion of p-uniform convexity from the theory of Banach spaces has been proposed as a generalization of the Alexandrov-Toponogov comparison theorems to Finsler manifolds. We prove that a natural Finsler metric on a strictly convex divisible domain is $β$-uniformly convex, where the exact constant $β$ is related to the regularity of the boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_20071 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $β$-Uniform Convexity and Divisible Domains Pompilio, Amelia Metric Geometry Differential Geometry Divisible convex sets have long been important in the study of Hilbert geometries. When a divisible convex set is an ellipsoid, the Hilbert geometry it induces is the hyperbolic space. In general, strictly convex divisible domains exhibit negative curvature properties, but only the ellipsoid is a CAT(0) space. The notion of p-uniform convexity from the theory of Banach spaces has been proposed as a generalization of the Alexandrov-Toponogov comparison theorems to Finsler manifolds. We prove that a natural Finsler metric on a strictly convex divisible domain is $β$-uniformly convex, where the exact constant $β$ is related to the regularity of the boundary. |
| title | $β$-Uniform Convexity and Divisible Domains |
| topic | Metric Geometry Differential Geometry |
| url | https://arxiv.org/abs/2410.20071 |