Weak Finsler structures and the Funk weak metric
By Athanase Papadopoulos, Marc Troyanov. Mathematical Proceedings of the Cambridge Philosophical Society, 2008
46 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/c58d6e802b5e807332cc275c2f5b0f134290bbdd
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Birkhoff's version of Hilbert's metric and its applications in analysis (2013, background): “Remark2.3. From equation (2.4) it follows that log kx−y′k ky−y′k = logM(x/y) for all x6= yin ΩC. The function FC : C◦ × C◦ → R given by FC(x,y) = logM(x/y) is called the Funk (weak) metric on C◦, see [56, 68]. Remark 2.4. Birkhoff’s version allows one to give a natural definition of Hilbert’s metric o”