Papadopoulos, AthanaseTroyanov, Marc2009-07-082009-07-082009-07-08200910.1017/S0305004109002461https://infoscience.epfl.ch/handle/20.500.14299/41222WOS:000269277400011We 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.text::journal::journal article::research article