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Proceedings of the American Mathematical Society

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

 
 

 

Structure of the efficient point set


Author: Đinh The Lục
Journal: Proc. Amer. Math. Soc. 95 (1985), 433-440
MSC: Primary 49A50; Secondary 90A14, 90C31
DOI: https://doi.org/10.1090/S0002-9939-1985-0806083-0
MathSciNet review: 806083
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Abstract: Let $C$ be a nontrivial cone and $X$ be a set in the $n$-dimensional Euclidean space. Denote by ${\text {E}}(X|C)$ the set of all efficient points of $X$ with respect to $C$. It will be proven that under some adequate assumptions ${\text {E}}(X|C)$ is homeomorphic to a simplex while $n = 2$, and for $n > 2$ it is a contractible set. Furthermore, the set of all weak efficient points of $X$ with respect to $C$ is arcwise connected and its local contractibility is equivalent to being a retract of $X$. The results presented in this study cover all topological properties of the efficient point set which have been obtained by Peleg and Morozov for the case when $C$ is the nonnegative orthant.


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Article copyright: © Copyright 1985 American Mathematical Society