Weak Finsler Structures and the Funk Metric

Athanase Papadopoulos (1) , Marc Troyanov (2)

1 IRMA - Institut de Recherche Mathématique Avancée
2 Section de Mathematiques [Lausanne]

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 and of convexity properties of the Funk weak metric.

Mots clés: Finsler structure; Funk weak metric; weak metric; Funk weak metric.

Domaines: Topologie géométrique [math.GT]

Athanase Papadopoulos, Marc Troyanov. Weak Finsler Structures and the Funk Metric. Mathematical Proceedings of the Cambridge Philosophical Society, 2009, 147 (2), ⟨10.1017/S0305004109002461⟩. ⟨hal-00276958⟩