From Funk to Hilbert Geometry

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.

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