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Journal of Geometry

, Volume 83, Issue 1–2, pp 22–31 | Cite as

Hilbert metrics and Minkowski norms

  • Thomas FoertschEmail author
  • Anders Karlsson
Original Paper

Abstract.

It is shown that the Hilbert geometry (Dh 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.

Mathematics Subject Classification (2000).

Primary 51Kxx 53C60 

Key words.

Hilbert metric asymptotic geometry 

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Copyright information

© Birkhäuser Verlag, Basel 2005

Authors and Affiliations

  1. 1.Mathematisches InstitutRheinische Friedrich-Wilhelms-Universität BonnBonnGermany
  2. 2.Department of MathematicsRoyal Institute of TechnologyStockholmSweden

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