Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Infimum Principle

Authors: Wladyslaw Kulpa and Andrzej Szymanski
Journal: Proc. Amer. Math. Soc. 132 (2004), 203-210
MSC (1991): Primary 52A07, 54H25, 90A56; Secondary 90D06
Published electronically: May 28, 2003
MathSciNet review: 2021263
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We utilize the technique of dual sets to prove a theorem on the attainment of a simultaneous infimum by a compatible family of functions. Corollaries to the theorem include, among others, the von Neumann Minimax Principle and Nash's Equilibrium Theorem.

References [Enhancements On Off] (What's this?)

  • 1. K. Devlin, Mathematician Awarded Nobel Prize, FOCUS 14 (6) (1994), 1, 3.
  • 2. D. Gale, John Nash and the Nobel Prize, FOCUS 15 (2) (1995), 4.
  • 3. D. Gale, The law of supply and demand, Math. Scandinavia 3 (1955), 155-169. MR 17:985f
  • 4. James Dugundji and Andrzej Granas, Fixed point theory. I, Monografie Matematyczne [Mathematical Monographs], vol. 61, Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1982. MR 660439
  • 5. Władysław Kulpa, Convexity and the Brouwer fixed point theorem, Proceedings of the 12th Summer Conference on General Topology and its Applications (North Bay, ON, 1997), 1997, pp. 211–235. MR 1718875
  • 6. J. F. Nash, Equilibrium points in $n$-person games, Proc. Nat. Acad. Sci. USA 36 (1950), 48-49. MR 11:192c
  • 7. J. F. Nash, Non-cooperative games, Annals of Math. 54 (1951), 286-295. MR 13:261g
  • 8. Hukukane Nikaidô, Convex structures and economic theory, Mathematics in Science and Engineering, Vol. 51, Academic Press, New York-London, 1968. MR 0277233
  • 9. H. Nikaido, On the classical multilateral exchange problem, Metroeconomics 8 (1956), Fas. 2. MR 18:266a
  • 10. H. Reitberger, Vietoris-Beglesches Abbildungstheorem, Vietoris-Lefschetz-Eilenberg-Montgomery-Beglescher Fixpunktsatz und Wirtschaftsnobelpreise, Jahresber. Deutsch. Math.-Verein. 103 (2001), no. 3, 67–73 (German). MR 1873323
  • 11. J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, John Wiley & Sons, New York, 1964.
  • 12. N. N. Vorob′ev, Game theory, Springer-Verlag, New York-Berlin, 1977. Lectures for economists and systems scientists; Translated and supplemented by S. Kotz; Applications of Mathematics, Vol. 7. MR 0452473

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 52A07, 54H25, 90A56, 90D06

Retrieve articles in all journals with MSC (1991): 52A07, 54H25, 90A56, 90D06

Additional Information

Wladyslaw Kulpa
Affiliation: Institute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland

Andrzej Szymanski
Affiliation: Department of Mathematics, Slippery Rock University, Slippery Rock, Pennsylvania 16057

Keywords: Simplicial space, dual family, fixed point
Received by editor(s): November 14, 2001
Received by editor(s) in revised form: August 14, 2002
Published electronically: May 28, 2003
Communicated by: Alan Dow
Article copyright: © Copyright 2003 American Mathematical Society