β-Uniform Convexity and Divisible Domains

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(-1) space. The notion of p-uniform convexity from the theory of Banach spaces has been proposed as a generalization of the Alexandrov-Toponogov comparison property 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.