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*, , is contractible. Furthermore, if 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.

**[1]**K. J. Arrow, E. W. Barankin, and D. Blackwell,*Admissible points of convex sets*, Contributions to the theory of games, vol. 2, Annals of Mathematics Studies, no. 28, Princeton University Press, Princeton, N. J., 1953, pp. 87–91. MR**0054919****[2]**David Gale,*The theory of linear economic models*, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1960. MR**0115801****[3]**Tjalling C. Koopmans,*Analysis of production as an efficient combination of activities*, Activity Analysis of Production and Allocation, Cowles Commission Monograph No. 13, John Wiley & Sons, Inc., New York, N. Y.; Chapman & Hall, Ltd., London, 1951, pp. 33–97. MR**0046017****[4]**M. Kurz and M. Majumdar,*Efficiency prices in infinite dimensional spaces*:*A synthesis*(to appear).**[5]**Mukul Majumdar,*Some approximation theorems on efficiency prices for infinite programs*, J. Econom. Theory**2**(1970), 399–410. MR**0449549****[6]**Hukukane Nikaidô,*Convex structures and economic theory*, Mathematics in Science and Engineering, Vol. 51, Academic Press, New York-London, 1968. MR**0277233****[7]**Bezalel Peleg,*Efficiency prices for optimal consumption plans*, J. Math. Anal. Appl.**29**(1970), 83–90. MR**0260408****[8]**Bezalel Peleg,*Efficiency prices for optimal consumption plans. II*, Israel J. Math.**9**(1971), 222–234. MR**0277234****[9]**Roy Radner,*A note on maximal points of convex sets in 𝑙_{∞}*, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 351–354. MR**0216276**

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

DOI:
https://doi.org/10.1090/S0002-9939-1972-0303916-2

Article copyright:
© Copyright 1972
American Mathematical Society