From Funk to Hilbert Geometry

2014/06/26 by Athanase Papadopoulos, Marc Troyanov, Papadopoulos, Athanase +1
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #Metric Geometry (math.MG) #math.GT #math.MG

paper · pdf · doi:10.48550/arxiv.1406.6983

To appear in the Handbook of Hilbert geometry (ed. A. Papadopoulos and M. Troyanov), European Mathematical Society, Zürich, 2014

arxiv created 2014/06/26 · arxiv updated 2014/06/27

Abstract

We survey some basic geometric properties of the Funk metric of a convex set in ℝⁿ. 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.

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