$\beta$-Uniform Convexity and Divisible Domains

TL;DR

This paper establishes that strictly convex divisible domains in Hilbert geometry are $eta$-uniformly convex with a constant linked to boundary regularity, extending curvature comparison concepts to Finsler geometries.

Contribution

It introduces $eta$-uniform convexity for Finsler metrics on divisible domains, generalizing curvature comparison theorems beyond ellipsoids.

Findings

Strictly convex divisible domains are $eta$-uniformly convex.

The $eta$ constant relates to boundary regularity.

Ellipsoids are the unique CAT(0) divisible domains.

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.