Hilbert metrics and Minkowski norms

Details Hilbert metrics and Minkowski norms Hilbert metrics and Minkowski norms

2004-07-12

arxiv.org/abs/math/0407197v1

Mathematics

Metric Geometry

11 pages, 3 figures

Scientific paper

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 $(V,||\cdot ||)$ 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.

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