Abstract

Given a metric space (X,d)(X,d) and a subset K⊆XK\subseteq X we say KK is dd-convex if for every x,y∈Kx,y\in K, the segment between them defined as
[x,y]:={z∈X:d(x,y)=d(x,z)+d(z,y)}[x,y]:=\{z\in X: d(x,y)=d(x,z)+d(z,y)\}
satisfy [x,y]⊆K[x,y]\subseteq K. We generalize this notion to subsets where this condition is satisfied for a subset of segments that cover the subset. Then we show versions of a Ky Fan's Lemma on spaces with this property. As an application, we introduce an approximation to the Goldbach's problem.

Author information

Contact details are reproduced from the original publication and may be historical.

Orlando Galdames-Bravo

Departament de Matemàtiques, CIPFP Vicente Blasco Ibáñez, València, Spain

galdames@uv.es

Keywords

  • Metric space
  • d-convexity
  • Ky Fan's Lemma
  • Goldbach's problem

Mathematics Subject Classification

52A40; 11P32

References

  1. H. Ben-El-Mechaiekh, S. Chebbi, M. Florenzano, J. V. Llinares: Abstract convexity and fixed points, J. Math. Anal. Appl. 222/1 (1998) 138–150.https://doi.org/10.1006/jmaa.1998.5918Google Scholar
  2. I. Bula: Strictly convex metric spaces and fixed points, Math. Moravica 3 (1999) 5–16.Google Scholar
  3. J. de Groot: Some special metrics in general topology, Colloq. Math. 6 (1958) 283–286.https://doi.org/10.4064/cm-6-1-283-286Google Scholar
  4. J. Diestel, H. Jarchow, A. Tonge: Absolutely Summing Operators, Cambridge University Press, Cambridge (1995).https://doi.org/10.1017/CBO9780511526138Google Scholar
  5. D. A. Edwards: The structure of superspace, Stud. Topol., Proc. Conf. Charlotte, N.C. (1975) 121–133.https://doi.org/10.1016/B978-0-12-663450-1.50017-7Google Scholar
  6. K. Fan: Minimax theorems, Proc. Natl. Acad. Sci. USA 39 (1953) 42–47.https://doi.org/10.1073/pnas.39.1.42Google Scholar
  7. T. Foertsch: Ball versus distance convexity of metric spaces, Beitr. Algebra Geometrie 45/2 (2004) 481–500.Google Scholar
  8. G. Giordano: The generalization and proof of Bertrand’s Postulate, Int. J. Math. Math. Sci. 10 (1987) 821–824.https://doi.org/10.1155/S0161171287000917Google Scholar
  9. M. Gromov: Structures Métriques pour les Variétés Riemanniennes, Textes Mathématiques Vol. 1, CEDIC/Fernand Nathan (1981).Google Scholar
  10. M. A. Khamsi: KKM and Ky Fan Theorems in hyperconvex metric spaces, J. Math. Analysis Appl. 204/1 (1996) 298–306.https://doi.org/10.1006/jmaa.1996.0438Google Scholar
  11. M. A. Khamsi, W. A. Kirk: An Introduction to Metric Spaces and Fixed Point Theory, John Wiley & Sons, New York (2001).https://doi.org/10.1002/9781118033074Google Scholar
  12. K. Menger: Untersuchungen über allgemeine Metrik. Untersuchungen I–III, Math. Ann. 100 (1928) 75–163.https://doi.org/10.1007/BF01448840Google Scholar
  13. S. Okada, W. J. Ricker, E. A. Sánchez-Pérez: Optimal Domain and Integral Extension of Operators Acting in Function Spaces, Operator Theory: Advances and Applications Vol. 180, Birkhäuser, Basel (2008).https://doi.org/10.1007/978-3-7643-8648-1Google Scholar
  14. A. Pietsch: Operator Ideals, North Holland, Amsterdam (1980).Google Scholar
  15. W. Rudin: Functional Analysis, McGraw-Hill, New York (1973).Google Scholar
  16. W. Sierpiński: Cardinal and Ordinal Numbers, Państwowe Wydawnictwa Naukowe, Warszawa (1958).Google Scholar
  17. P. S. Soltan: Helly’s theorem for d-convex sets (Russian), Dokl. Akad. Nauk SSSR Ser. Mat. 205/3 (1972) 537–539; English translation in Soviet Math. Dokl. 13 (1972) 975–978.Google Scholar
  18. V. P. Soltan, P. S. Soltan: d-Convex functions (Russian), Dokl. Akad. Nauk SSSR Ser. Mat. 249/5 (1979) 555–558; English translation in Soviet Math. Dokl. 20 (1979) 1323–1326.https://doi.org/10.1016/S0040-4039(01)86138-4Google Scholar
  19. K. T. Wen: A new KKM theorem in L-convex spaces and some applications, Comput. Math. Appl. 56/11 (2008) 2781–2785.https://doi.org/10.1016/j.camwa.2008.07.014Google Scholar

Suggested citation

O. Galdames-Bravo. “Ky Fan's Lemma for Metric Spaces and an Approximation to the Goldbach's Problem.” Journal of Convex Analysis 31 (2024), No. 1, 265–277. https://doi.org/10.68381/jca31016

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2024. Rights now held by Banach Press LLC.