Hilbert metrics and Minkowski norms

Abstract.

It is shown that the Hilbert geometry (D, h _D ) associated to a bounded convex domain $D \subset \mathbb{E}^{n}$ is isometric to a normed vector space ${\left( {{\text{V}},{\left\| {\, \cdot \,} \right\|}} \right)}$ if and only if D is an open n-simplex. One further result on the asymptotic geometry of Hilbert’s metric is obtained with corollaries for the behavior of geodesics. Finally we prove that every geodesic ray in a Hilbert geometry converges to a point of the boundary.