Chapitre D'ouvrage Année : 2014

From Funk to Hilbert Geometry

Résumé

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.

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hal-01015584 , version 1 (26-06-2014)

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Athanase Papadopoulos, Marc Troyanov. From Funk to Hilbert Geometry. Handbook of Hilbert geometry, 22, European Mathematical Society Publishing House, p. 33-68., 2014, IRMA Lectures in Mathematics and Theoretical Physics, 978-3-03719-147-7. ⟨10.4171/147-1/2⟩. ⟨hal-01015584⟩
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