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Weak Finsler structures and the Funk weak metric


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.

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[2] J. C. Álvarez Paiva Some problems in Finsler geometry, in: Handbook of Differential Geometry. Vol. II, 133 (Elsevier, 2006).

[4] D. Bao , S. S. Chern and Z. Shen An introduction to Riemann-Finsler geometry. Graduate Texts in Mathematics (Springer Verlag, 2000).

[7] H. Busemann Local metric geometry. Trans. Amer. Math. Soc. 56 (1944), 200274.

[12] P. Funk Über geometrien, bei denen die geraden die kürzesten sind. Math. Ann. 101 (1929), 226237.

[17] A. Papadopoulos and M. Troyanov Harmonic symmetrization of convex sets and of Finsler structures, with applications to Hilbert geometry. Expo. Math. 27 (2009), 109124.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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