Handbook of Hilbert Geometry

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pp: 33–67

DOI: 10.4171/147-1/2

From Funk to Hilbert geometry

Athanase Papadopoulos[1] and Marc Troyanov[2]

(1) Université de Strasbourg, France
(2) Ecole Polytechnique Fédérale de Lausanne, Switzerland

We survey some basic geometric properties of the Funk metric of a convex set in $\mathbb{R}^n$. 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.

Keywords: Funk metric, convexity, Hilbert metric, Busemann's methods