Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-26T14:48:54.089Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak Finsler structures and the Funk weak metric

Published online by Cambridge University Press:  01 September 2009

ATHANASE PAPADOPOULOS  and
MARC TROYANOV
ATHANASE PAPADOPOULOS
Affiliation:
Institut de Recherche Mathématique Avancée, Université de Strasbourg and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France. e-mail: [email protected]
MARC TROYANOV
Affiliation:
Section de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 2 , September 2009 , pp. 419 - 437
Copyright
Copyright © Cambridge Philosophical Society 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Álvarez Paiva, J. C. and Durán, C.An Introduction to Finsler Geometry (Publicaciones de la Escuela Venezolana de Matematicas, 1998).Google Scholar
[2]Álvarez Paiva, J. C. Some problems in Finsler geometry, in: Handbook of Differential Geometry. Vol. II, 1–33 (Elsevier, 2006).Google Scholar
[3]Bao, D., Bryant, R. L., Chern, S. S. and Shen, Z. (editors). A Sampler of Finsler Geometry, MSRI Publications 50 (Cambridge University Press, 2004).Google Scholar
[4]Bao, D., Chern, S. S. and Shen, Z.An introduction to Riemann-Finsler geometry. Graduate Texts in Mathematics (Springer Verlag, 2000).CrossRefGoogle Scholar
[5]Bao, D., Robles, C. and Shen, Z.Zermelo navigation on Riemannian manifolds. J. Diff. Geom. 66 (2004), 377–435.Google Scholar
[6]Busemann, H. Metric methods in Finsler spaces and in the foundations of geometry. Ann. Math. Stud. 8 (Princeton University Press, 1942).Google Scholar
[7]Busemann, H.Local metric geometry. Trans. Amer. Math. Soc. 56 (1944), 200–274.CrossRefGoogle Scholar
[8]Busemann, H.The Geometry of Geodesics (Academic Press, 1955) (reprinted by Dover, 2005).Google Scholar
[9]Busemann, H.Recent synthetic differential geometry. Ergeb. Math. Grenzgeb. 54 (1970).Google Scholar
[10]Chern, S. S. and Shen, Z. Riemann-Finsler geometry. Nankai Tracts Math. (2005).CrossRefGoogle Scholar
[11]Eggleston, H. G. Convexity. Cambridge Tracts in Mathematics and Mathematical Physics No. 47 (Cambridge University Press, 1958).Google Scholar
[12]Funk, P.Über geometrien, bei denen die geraden die kürzesten sind. Math. Ann. 101 (1929), 226–237.Google Scholar
[13]Hausdorff, F.Set Theory (Chelsea, 1957) translation of Grundzüge der Mengenlehre, first ed. 1914.Google Scholar
[14]Hilbert, D.Grundlagen der Geometrie. (B. G. Teubner, 1899), (several later editions revised by the author, and several translations).Google Scholar
[15]Minkowski, H. Theorie der konvexen Körper, insbesondere Begründung ihres Ober-flächenbegriffs. in Gesammelte Abhandlungen (Teubner, Leipzig, 1911).Google Scholar
[16]Papadopoulos, A. and Troyanov, M.Weak metrics on Euclidean domains. JP Journal of Geometry and Topology. Volume 7, Issue 1 (March 2007), pp. 23–44.Google Scholar
[17]Papadopoulos, A. and Troyanov, M.Harmonic symmetrization of convex sets and of Finsler structures, with applications to Hilbert geometry. Expo. Math. 27 (2009), 109–124.CrossRefGoogle Scholar
[18]Papadopoulos, A. and Troyanov, M. Symmetrization of convex sets and applications. in preparartion.Google Scholar
[19]Ribeiro, H.Sur les espaces à métrique faible. Port. Math. 4 (1943), 21–40.Google Scholar
[20]Thompson, A. C.Minkowski Geometry. Encyclopedia of Mathematics and its Applications, 63 (Cambridge University Press, 1996).Google Scholar
[21]Webster, R.Convexity (Oxford University Press, 1994).Google Scholar
[22]Zaustinsky, E. M.Spaces with nonsymmetric distance. Mem. Amer. Math. Soc. No. 34 (1959).Google Scholar