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.
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Foertsch, T., Karlsson, A. Hilbert metrics and Minkowski norms. J. geom. 83, 22–31 (2005). https://doi.org/10.1007/s00022-005-0005-1
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DOI: https://doi.org/10.1007/s00022-005-0005-1