Hilbert metrics and Minkowski norms

Abstract

It is shown that the Hilbert geometry (D, h D ) associated to a bounded convex domain D ⊂ E n 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.

Horoballs in simplices and Minkowski spaces

“…Note that it is well known that (Δ,k) is isometrically isomorphic to a certain Minkowski space with polyhedral unit ball (see [16,Proposition 1.7], [6,Proposition 7], or [7]). Therefore, certain techniques from polyhedral Minkowski spaces can be used for (Δ,k).…”

Hilbert domains that admit a quasi-isometric embedding into Euclidean space

“…The proof of Theorem 1.4 is based on an idea developed by Foertsch and Karlsson in their paper [10]. For this purpose, let us first introduce a definition:…”

Volume entropy of Hilbert geometries

“…He constructed a special class of examples, now called Hilbert geometries [Hilbert 1895;, which have since attracted much interest; see, for example, [Nasu 1961;de la Harpe 1993;Karlsson and Noskov 2002;Socié-Méthou 2004;Foertsch and Karlsson 2005;Benoist 2006; Colbois and Vernicos 2007], and the two complementary surveys [Benoist 2008] and [Vernicos 2005].…”