Weak Finsler structures and the Funk weak metric

Source abstract

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.