From Funk to Hilbert geometry

  • Athanase Papadopoulos

    Université de Strasbourg, France
  • Marc Troyanov

    Ecole Polytechnique Fédérale de Lausanne, Switzerland
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We survey some basic geometric properties of the Funk metric of a convex set in . In particular, we study its geodesics, its topology, its metric balls, its convexity properties, its perpendicularity theory and its isometries. The Hilbert metric is a symmetrization of the Funk metric, and we show some properties of the Hilbert metric that follow directly from the properties we prove for the Funk metric.