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Weak Finsler structures and the Funk weak metric

Published online by Cambridge University Press:  01 September 2009

ATHANASE PAPADOPOULOS  and
MARC TROYANOV
ATHANASE PAPADOPOULOS
Affiliation:
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
MARC TROYANOV
Affiliation:
Section de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland. e-mail: marc.troyanov@epfl.ch
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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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 2 , September 2009 , pp. 419 - 437
Copyright
Copyright © Cambridge Philosophical Society 2009

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