On Non-positive Curvature Properties of the Hilbert Metric

Abstract

In this paper, we consider different types of non-positive curvature properties of the Hilbert metric of a convex domain in $${\mathbb {R}^n}$$ . 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.