Missouri Journal of Mathematical Sciences

Intersection Theorems for Closed Convex Sets and Applications

Hichem Ben-El-Mechaiekh

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Abstract

A number of landmark existence theorems of nonlinear functional analysis follow in a simple and direct way from the basic separation of convex closed sets in finite dimension via elementary versions of the Knaster-Kuratowski-Mazurkiewicz principle - which we extend to arbitrary topological vector spaces - and a coincidence property for so-called von Neumann relations. The method avoids the use of deeper results of topological essence such as the Brouwer Fixed Point Theorem or the Sperner's Lemma and underlines the crucial role played by convexity. It turns out that the convex KKM Principle is equivalent to the Hahn-Banach Theorem, the Markov-Kakutani Fixed Point Theorem, and the Sion-von Neumann Minimax Principle.

Article information

Source
Missouri J. Math. Sci., Volume 27, Issue 1 (2015), 47-63.

Dates
First available in Project Euclid: 3 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.mjms/1449161367

Mathematical Reviews number (MathSciNet)
MR3431115

Zentralblatt MATH identifier
1341.52002

Subjects
Primary: 52A07: Convex sets in topological vector spaces [See also 46A55]
Secondary: 32F32: Analytical consequences of geometric convexity (vanishing theorems, etc.) 32F27: Topological consequences of geometric convexity 47H04: Set-valued operators [See also 28B20, 54C60, 58C06] 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30] 47N10: Applications in optimization, convex analysis, mathematical programming, economics

Keywords
Separation of convex sets intersection theorems convex KKM Theorem fixed points for von Neumann relations coincidence systems of nonlinear inequalities variational inequalities minimization of functionals Markov-Kakutani Fixed Point Theorem Hahn-Banach theorem

Citation

Ben-El-Mechaiekh, Hichem. Intersection Theorems for Closed Convex Sets and Applications. Missouri J. Math. Sci. 27 (2015), no. 1, 47--63. https://projecteuclid.org/euclid.mjms/1449161367


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References

  • H. Ben-El-Mechaiekh and R. W. Dimand, A simpler proof of the Von Neumann minimax theorem, Amer. Math. Monthly, 118 (2011), 636–641.
  • H. Ben-El-Mechaiekh, P. Deguire, and A. Granas, Points fixes et coincidences pour les fonctions multivoques I (applications de Ky Fan), C. R. Acad. Sci. Paris, 295 (1982), 337–340.
  • H. Ben-El-Mechaiekh, P. Deguire, and A. Granas, Points fixes et coincidences pour les fonctions multivoques II (applications de type $\Phi $ et $\Phi ^{\ast })$, C. R. Acad. Sci. Paris, 295 (1982), 381–384.
  • H. Ben-El-Mechaiekh, P. Deguire, and A. Granas, Points fixes et coincidences pour les fonctions multivoques III (applications de type $M$ et $M^{\ast })$, C. R. Acad. Sc. Paris, 305 (1987), 381–384.
  • C. Berge, Sur une propriété combinatoire des ensembles convexes, C. R. Acad. Sc. Paris, 248 (1959), 2698–2699.
  • H. Brézis, Analyse Fonctionelle, Masson, Paris, 1983.
  • J. Dugundji and A. Granas, Fixed Point Theory, Vol. I, Monografie Matematycne 61, Warszawa, 1982.
  • A. Granas, KKM-Maps and their applications nonlinear problems, The Scottish Book, R. D. Mauldin, ed., Birkhauser, Massachusetts, 1981, 45–61.
  • A. Granas, Méthodes topologiques en analyse convexe, Séminaire de Mathématiques Supérieures 110, Les Presses de l'Université de Montréal, 1990.
  • A. Granas and M. Lassonde, Some Elementary General Principles of Convex Analysis, Top. Meth. in Nonlin. Anal., 5 (1995), 23–37.
  • A. Ghouila-Houri, Sur l'étude combinatoire des familles de convexes, C. R. Acad. Sc. Paris, 252 (1961), 494–496.
  • K. Fan, A Generalization of Tychonoff's fixed point theorem, Mat. Ann., 142 (1961), 305–310.
  • K. Fan, Covering properties of convex sets and fixed point theorems in topological vector spaces, Symposium on Infinite Dimensional Topology, R. D. Anderson, ed., Annals of Mathematics Studies, Princeton University Press, 1972, 79–92.
  • K. Fan, Some properties of convex sets related to fixed point theorems, Mat. Ann., 266 (1984), 519–537.
  • C. D. Horvath and M. Lassonde, Intersection of sets with n-connected unions, Proc. Amer. Math. Soc., 125 (1997), 1209–1214.
  • S. Kakutani, A proof of the Hahn-Banach theorem via a fixed point theorem, Selected Papers, Vol. I, Birkhäuser, 1986, 154–158.
  • V. L. Klee, On certain intersection properties of convex sets, Canad. J. Math., 3 (1951), 272–275.
  • B. Knaster, C. Kuratowski, and S. Mazurkiewicz, Ein beweis des fixpunktsatzes für n-dimensionale simplexe, Fund. Math., 14 (1929), 132–138.
  • M. Magill and M. Quinzii, Theory of Incomplete Markets, MIT Press, Cambridge, MA, 1996.
  • S. Park, A brief history of the KKM theory, RIMS Kô kyûroku, Kyoto Univ., 1643 (2009), 1–16.
  • D. Werner, A proof of the Markov-Kakutani fixed point theorem via the Hahn-Banach theorem, Extracta Mathematicae, 8 (1992), 37–38.
  • G. X.-Z. Yuan, KKM Theory and Applications in Nonlinear Analysis, Vol. 218 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, 1999.