Hilbert metrics and Minkowski norms

By T. Foertsch, Anders Karlsson. 2004

49 citations recorded by the source corpus.

Abstract.It is shown that the Hilbert geometry (D, hD) 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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/e9a6d3d44cb9e9e372513bff732d7e4886e4b57a

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In Applied Mathematics

1. David Dumas
2. Roger D. Nussbaum
3. Bas Lemmens
4. Michael Wolf
5. Jean-Marc Schlenker

Citations in the corpus

• Birkhoff's version of Hilbert's metric and its applications in analysis (2013, background): “In [21] Foertsch and Karlsson showed that the only Hilbert geometry isometric to a normed space is the one on an open simplex.”
• Polynomial cubic differentials and convex polygons in the projective plane (2014, background): “in [dlH93] [FK05] [Ber09] [CV11] [CVV11]), one might ask whether such results could be brought to bear on the study of polygonal affine spheres.”