ArticleOriginal scientific text
Title
Minimax theorems with applications to convex metric spaces
Authors 1
Affiliations
- Fachbereich Mathematik, Technische Hochschule Darmstadt, Schlossgartenstr. 7, D-64289 Darmstadt, Germany
Abstract
A minimax theorem is proved which contains a recent result of Pinelis and a version of the classical minimax theorem of Ky Fan as special cases. Some applications to the theory of convex metric spaces (farthest points, rendez-vous value) are presented.
Bibliography
- A. A. Astaneh, On singletonness of uniquely remotal sets, Indian J. Pure Appl. Math. 17 (9) (1986), 1137-1139.
- R. G. Bilyeu, Metric definition of the linear structure, Proc. Amer. Math. Soc. 25 (1970), 205-206.
- L. M. Blumenthal and K. Menger, Studies in Geometry, Freeman, San Francisco, 1970.
- J. Cleary, S. A. Morris and D. Yost, Numerical geometry-numbers for shapes, Amer. Math. Monthly 93 (1986), 260-275.
- M. De Wilde, Doubles limites ordonnées et théorèmes de minimax, Ann. Inst. Fourier (Grenoble) 24 (1974), 181-188.
- K. Fan, Minimax theorems, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 42-47.
- I. Glicksberg, Minimax theorem for upper and lower semi-continuous payoffs, The RAND Corporation Research Memorandum RM-478 (1950).
- O. Gross, The rendezvous value of a metric space, in: Advances in Game Theory, Ann. of Math. Stud. 52, Princeton Univ. Press, 1964, 49-53.
- Y. Kijima, Fixed points of nonexpansive self-maps of a compact metric space, J. Math. Anal. Appl. 123 (1987), 114-116.
- J. Kindler, Minimaxtheoreme und das Integraldarstellungsproblem, Manuscripta Math. 29 (1979), 277-294.
- J. Kindler, Minimaxtheoreme für die diskrete gemischte Erweiterung von Spielen und ein Approximationssatz, Math. Operationsforsch. Statist. Ser. Optim. 11 (1980), 473-485.
- J. Kindler, Minimax theorems with one-sided randomization, Acta Math. Hungar., to appear.
- H. König, Über das von Neumannsche Minimax-Theorem, Arch. Math. (Basel) 19 (1968), 482-487.
- S. A. Morris and P. Nickolas, On the average distance property of compact connected metric spaces, Arch. Math. (Basel) 40 (1983), 459-463.
- M. Neumann, Bemerkungen zum von Neumannschen Minimaxtheorem, ibid. 29 (1977), 96-105.
- J. E. L. Peck and A. L. Dulmage, Games on a compact set, Canad. J. Math. 9 (1957), 450-458.
- I. F. Pinelis, On minimax risk, Theory Probab. Appl. 35 (1990), 104-109.
- I. F. Pinelis, On minimax estimation of regression, ibid., 500-512.
- W. Stadje, A property of compact connected spaces, Arch. Math. (Basel) 36 (1981), 275-280.
- W. Takahashi, A convexity in metric space and nonexpansive mappings, I, Kōdai Math. Sem. Rep. 22 (1970), 142-149.
- L. Yang and J. Zhang, Average distance constants of some compact convex space, J. China Univ. Sci. Tech. 17 (1987), 17-23.