Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Weak Finsler structures and the Funk weak metric

ATHANASE PAPADOPOULOSa1 and MARC TROYANOVa2

a1 Institut de Recherche Mathématique Avancée, Université de Strasbourg and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France. e-mail: papadopoulos@math.u-strasbg.fr

a2 Section de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland. e-mail: marc.troyanov@epfl.ch

Abstract

We discuss general notions of metrics and of Finsler structures which we call weak metrics and weak Finsler structures. Any convex domain carries a canonical weak Finsler structure, which we call its tautological weak Finsler structure. We compute distances in the tautological weak Finsler structure of a domain and we show that these are given by the so-called Funk weak metric. We conclude the paper with a discussion of geodesics, of metric balls, of convexity, and of rigidity properties of the Funk weak metric.

(Received April 05 2008)

(Revised December 23 2008)