Hilbert metrics and Minkowski norms

Original · EN

It is shown that the Hilbert geometry (D,hD) associated to a bounded convex domain D⊂ Eⁿ is isometric to a normed vector space (V,||· ||) 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.