Weak Finsler Strutures and the Funk Metric

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 and of convexity properties of the Funk weak metric.

18 pages, 3 figures

Cited by in corpus (7)

• A maximin characterization of the escape rate of nonexpansive mappings in metrically convex spaces
• Plateau's problem in Finsler 3-space
• Understanding and monitoring the evolution of the Covid-19 epidemic from medical emergency calls: the example of the Paris area
• Convex bodies in Euclidean and Weil-Petersson geometries
• Timelike Hilbert and Funk geometries
• Firm non-expansive mappings in weak metric spaces
• On different curvatures of spheres in Funk geometry