From Funk to Hilbert Geometry

TL;DR

This paper surveys the geometric properties of the Funk and Hilbert metrics on convex sets in Euclidean space, focusing on geodesics, topology, convexity, and isometries, and explores their interrelations.

Contribution

It provides a comprehensive analysis of the Funk metric's properties and derives new insights into the Hilbert metric as a symmetrization of Funk, highlighting their geometric features.

Findings

Characterization of geodesics in Funk metric

Topological properties of the Funk metric space

Relations between Funk and Hilbert metrics

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.