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arXiv · 2410.20071

$\beta$-Uniform Convexity and Divisible Domains

Amelia Pompilio
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Abstract

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 $\beta$-uniformly convex, where the exact constant $\beta$ is related to the regularity of the boundary.

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Amelia Pompilio. 2024-10-26. $\beta$-Uniform Convexity and Divisible Domains. https://arxiv.org/abs/2410.20071

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