Sur un principe géométrique en analyse convexe

Andrzej Granas; Marc Lassonde

Studia Mathematica (1991)

  • Volume: 101, Issue: 1, page 1-18
  • ISSN: 0039-3223

Abstract

top
In this note we present we present a new elementary approach in the theory of minimax inequalities. The proof of the main result (called the geometric principle) uses only some simple properties of convex functions. The geometric principle (which is equivalent to the well-known lemma of Klee [13]) is shown to have numerous applications in different areas of mathematics.

How to cite

top

Granas, Andrzej, and Lassonde, Marc. "Sur un principe géométrique en analyse convexe." Studia Mathematica 101.1 (1991): 1-18. <http://eudml.org/doc/215891>.

@article{Granas1991,
author = {Granas, Andrzej, Lassonde, Marc},
journal = {Studia Mathematica},
keywords = {KKM principle; minimax inequalities; geometric principle},
language = {fre},
number = {1},
pages = {1-18},
title = {Sur un principe géométrique en analyse convexe},
url = {http://eudml.org/doc/215891},
volume = {101},
year = {1991},
}

TY - JOUR
AU - Granas, Andrzej
AU - Lassonde, Marc
TI - Sur un principe géométrique en analyse convexe
JO - Studia Mathematica
PY - 1991
VL - 101
IS - 1
SP - 1
EP - 18
LA - fre
KW - KKM principle; minimax inequalities; geometric principle
UR - http://eudml.org/doc/215891
ER -

References

top
  1. [1] H. Asakawa, Maximal monotone operators associated with saddle functions, TRU Math. 22 (2) (1986), 47-71. Zbl0687.90096
  2. [2] C. Berge, Espaces topologiques, Fonctions multivoques, Dunod, Paris 1959. Zbl0088.14703
  3. [3] H. F. Bohnenblust, S. Karlin and L. Shapley, Games with continuous pay-off, in: Ann. of Math. Stud. 24 (1950), 181-192 Zbl0168.41208
  4. [4] H. Debrunner und P. Flor, Ein Erweiterungssatz für monotone Mengen, Arch. Math. (Basel) 15 (1964), 445-447. Zbl0129.09203
  5. [5] J. Dugundji and A. Granas, Fixed Point Theory, Vol. I, Monografie Mat. 61, Polish Scientific Publishers, Warszawa 1982. 
  6. [6] K. Fan, Minimax theorems, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 42-47. 
  7. [7] K. Fan, Existence theorems and extreme solutions for inequalities concerning convex functions or linear transformations, Math. Z. 68 (1957), 205-216. Zbl0078.10204
  8. [8] K. Fan, A generalization of Tychonoff's fixed point theorem, Math. Ann. 142 (1961), 305-310. Zbl0093.36701
  9. [9] K. Fan, Extensions of two fixed point theorems of F. E. Browder, Math. Z. 112 (1969), 234-240. Zbl0185.39503
  10. [10] K. Fan, A minimax inequality and applications, in: Inequalities III, O. Shisha (ed.), Academic Press, New York 1972, 103-113. 
  11. [11] K. Fan, I. Glicksberg and A. J. Hoffman, Systems of inequalities involving convex functions, Proc. Amer. Math. Soc. 8 (1957), 617-622. Zbl0079.02002
  12. [12] B. Halpern and G. Bergman, A fixed point theorem for inward and outward maps, Trans. Amer. Math. Soc. 130 (1968), 353-358. Zbl0153.45602
  13. [13] V. L. Klee, On certain intersection properties of convex sets, Canad. J. Math. 3 (1951), 272-275. Zbl0042.40701
  14. [14] B. Knaster, C. Kuratowski und S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für n-dimensionale Simplexe, Fund. Math. 14 (1929), 132-137. Zbl55.0972.01
  15. [15] H. Kneser, Sur un théorème fondamental de la théorie des jeux, C. R. Acad. Sci. Paris 234 (1952), 2418-2420. Zbl0046.12201
  16. [16] H. König, Über das von Neumannsche minimax theorem, Arch. Math. (Basel) 19 (1968), 482-487. Zbl0179.21001
  17. [17] M. Lassonde, Multi-applications KKM en analyse non linéaire, Thèse de Doctorat, Université de Montréal, 1978. 
  18. [18] M. Neumann, Bemerkungen zum von Neumannschen Minimaxtheorem, Arch. Math. (Basel) 29 (1977), 96-105. 
  19. [19] H. Nikaidô, On von Neumann's minimax theorem, Pacific J. Math. 4 (1954), 65-72. Zbl0055.10004
  20. [20] M. Sion, On general minimax theorems, ibid. 8 (1958), 171-176. Zbl0081.11502
  21. [21] E. Sperner, Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes, Abh. Math. Sem. Univ. Hamburg 6 (1928), 265-272. Zbl54.0614.01
  22. [22] F. A. Valentine, Convex Sets, McGraw-Hill, New York 1964. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.