arXiv · 2204.04921
We extend Alberti's Rank-One Theorem to $\mathrm{RCD}(K,N)$ metric measure spaces.
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Gioacchino Antonelli, Camillo Brena, Enrico Pasqualetto. 2023-10-18. The Rank-One Theorem on $\mathrm{RCD}$ spaces. https://doi.org/10.2140/apde.2024.17.2797
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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 by Shin-Ichi Ohta 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 constant $β$ is related to the regularity of the boundary. We use this to show, with AI assistance, that the Hilbert metric, under suitable local and scale-dependent assumptions, is $β$-uniformly convex on such domains.
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