# From Funk to Hilbert geometry

Université de Strasbourg, France
• ### Marc Troyanov

Ecole Polytechnique Fédérale de Lausanne, Switzerland

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## Abstract

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.