  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.
