Abstract

It has been recently proved that balls in a proper geodesic Ptolemy space are strictly convex. In this work we further study Ptolemy spaces in this direction. In particular we prove the uniform convexity of geodesic Ptolemy spaces when assuming a uniform continuity condition on a midpoint map and use this property to deduce different geometrical and convexity related facts about such spaces. We also show that a nice modulus of uniform convexity can be found in such a setting

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Rafa Espínola

Dep. de Análisis Matemático - IMUS, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain

espinola@us.es

Adriana Nicolae

Dept. of Mathematics, Babeş-Bolyai University, Kogălniceanu 1, 400084 Cluj-Napoca, Romania

anicolae@math.ubbcluj.ro

References

  1. Y. Benyamini, J. Lindenstrauss: Geometric Nonlinear Functional Analysis, Vol. 1, Colloquium Publications 48, Amer. Math. Soc., Providence (2000).https://doi.org/10.1090/coll/048Google Scholar
  2. M. R. Bridson, A. Haefliger: Metric Spaces of Non-positive Curvature, Springer, Berlin (1999).https://doi.org/10.1007/978-3-662-12494-9Google Scholar
  3. S. M. Buckley, K. Falk, D. J. Wraith: Ptolemaic spaces and CAT(0), Glasgow Math. J. 51 (2009) 301–314.https://doi.org/10.1017/S0017089509004984Google Scholar
  4. D. Burago, Y. Burago, S. Ivanov: A Course in Metric Geometry, Graduate Studies in Math. 33, Amer. Math. Soc., Providence (2001).https://doi.org/10.1090/gsm/033Google Scholar
  5. R. Espínola, A. Fernández-León, B. Piątek: Fixed points of single- and set-valued mappings in uniformly convex metric spaces with no metric convexity, Fixed Point Theory Appl. 2010 (2010), Article ID 169837, 16 p.https://doi.org/10.1155/2010/169837Google Scholar
  6. R. Espínola, A. Nicolae: Geodesic Ptolemy spaces and fixed points, Nonlinear Anal. 74 (2011) 27–34.https://doi.org/10.1016/j.na.2010.08.009Google Scholar
  7. T. Foertsch, A. Lytchak, V. Schroeder: Nonpositive curvature and the Ptolemy inequality, Int. Math. Res. Not. 2007(22) (2007), Article ID rnm100, 15 p.Google Scholar
  8. T. Foertsch, V. Schroeder: Group actions on geodesic Ptolemy spaces, Trans. Amer. Math. Soc. 363 (2011) 2891–2906.https://doi.org/10.1090/S0002-9947-2011-05121-4Google Scholar
  9. T. Foertsch, V. Schroeder: Hyperbolicity, CAT(-1)-spaces and the Ptolemy inequality, Math. Ann. 350 (2011) 339–356.https://doi.org/10.1007/s00208-010-0560-0Google Scholar
  10. K. Goebel, S. Reich: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York (1984).Google Scholar
  11. U. Kohlenbach, L. Leuştean: Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces, J. Eur. Math. Soc. 12 (2010) 71–92.https://doi.org/10.4171/jems/190Google Scholar
  12. S. Reich, I. Shafrir: Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal. 15 (1990) 537–558.https://doi.org/10.1016/0362-546X(90)90058-OGoogle Scholar
  13. I. J. Schoenberg: A remark on M. M. Day’s characterization of inner-product spaces and a conjecture of L. M. Blumenthal, Proc. Amer. Math. Soc. 3 (1952) 961–964.https://doi.org/10.1090/S0002-9939-1952-0052035-9Google Scholar

Suggested citation

R. Espínola, A. Nicolae. “Uniform Convexity of Geodesic Ptolemy Spaces.” Journal of Convex Analysis 20 (2013), No. 3, 689–700. https://doi.org/10.68381/jca20039

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Published by Heldermann Verlag, 2013. Rights now held by Banach Press.