On non-positive curvature properties of the Hilbert metric

TL;DR

This paper explores the non-positive curvature properties of the Hilbert metric on convex domains, establishing equivalences among various concepts and demonstrating that certain conditions imply the domain is an ellipsoid with a Riemannian metric.

Contribution

It provides a survey of curvature concepts for the Hilbert metric and proves a rigidity result linking Berwald property to ellipsoids in convex domains.

Findings

Certain non-positive curvature properties are equivalent for the Hilbert metric.

If the Hilbert metric is Berwald, then the domain must be an ellipsoid.

The Hilbert metric reduces to a Riemannian metric on ellipsoids.

Abstract

In this paper, we consider different types of non-positive curvature properties of the Hilbert metric of a convex domain in R^n. First, we survey the relationships among the concepts and prove that in the case of Hilbert metric some of them are equivalent. Furthermore, we show some condition which implies the rigidity feature: if the Hilbert metric is Berwald, i.e., its Finslerian Chern connection reduces to a linear one, then the domain is an ellipsoid and the metric is Riemannian.