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Vietnam Journal of Mathematics 33:4 (2005) 463-468

 On Efficient Sets in R2

Hoang Xuan Phu

Abstract. Let A R2 be a nonempty closed convex subset and C R2 be a nonempty nontrivial convex cone. Due to Luc (1985 and 1989), if A is compact and if the closure  is pointed, then the efficient set E(A|C) of A w.r.t. C is homeomorphic to a nonempty closed interval of R1, whose proof was completed by Huy, Phuong, and Yen (2002). Huy (2003) extended this result by replacing the compactness of A with the compactness of A  ({a}- ), for all a  A. In this paper, we show the same conclusion in a much shorter way and under essentially weaker assumption, namely C is pointed and there exists an a  A such that A  ({a} - C) is bounded. Moreover, the weakly efficient set Ew(A|C) w.r.t. any convex cone C having nonempty interior is homeomorphic to a closed interval in R1 even if C is not pointed.

 

 

 

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