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)



Topological properties of the efficient point set

Author: Bezalel Peleg
Journal: Proc. Amer. Math. Soc. 35 (1972), 531-536
MSC: Primary 90A99; Secondary 54F05, 90D99
MathSciNet review: 0303916
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let y be a closed and convex subset of a Euclidean space. We prove that the set of efficient points of Y, $ M(Y)$, is contractible. Furthermore, if $ M(Y)$ is closed (compact) then it is a retract of a convex closed (compact) set. Our proof relies on the Arrow-Barankin-Blackwell Theorem. A new proof is supplied for that theorem.

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

  • [1] K. J. Arrow, E. W. Barankin and D. Blackwell, Admissible points of convex sets, Contributions to the Theory of Games, vol. 2, Ann. of Math. Studies, no. 28, Princeton Univ. Press, Princeton, N.J., 1953, pp. 87-91. MR 14, 998. MR 0054919 (14:998h)
  • [2] D. Gale, The theory of linear economic models, McGraw-Hill, New York, 1960. MR 22 #6599. MR 0115801 (22:6599)
  • [3] T. C. Koopmans, Analysis of production as an efficient combination of activities, Activity Analysis of Production and Allocation, Wiley, New York; Chapman and Hall, London, 1951. MR 13, 670. MR 0046017 (13:670b)
  • [4] M. Kurz and M. Majumdar, Efficiency prices in infinite dimensional spaces: A synthesis (to appear).
  • [5] M. Majumdar, Some approximation theorems on efficiency prices for infinite programs, J. Economic Theory 2 (1970), 399-410. MR 0449549 (56:7851)
  • [6] H. Nikaido, Convex structures and economic theory, Academic Press, New York, 1968. MR 0277233 (43:2970)
  • [7] B. Peleg, Efficiency prices for optimal consumption plans, J. Math. Anal. Appl. 29 (1970), 83-90. MR 41 #5034. MR 0260408 (41:5034)
  • [8] -, Efficiency prices for optimal consumption plans. II, Israel J. Math. 9 (1971), 222-234. MR 0277234 (43:2971)
  • [9] R. Radner, A note on maximal points of convex sets in $ {l_\infty }$, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), vol. 1: Statistics, Univ. of California Press, Berkeley, Calif., 1967, pp. 351-354. MR 35 #7111. MR 0216276 (35:7111)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 90A99, 54F05, 90D99

Retrieve articles in all journals with MSC: 90A99, 54F05, 90D99

Additional Information

Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society