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

On Efficient Sets in $\mathbb R^2$

Hoang Xuan Phu

Abstract. Let $A\subset \mathbb R^2$ be a nonempty closed convex subset and $C\subset \mathbb R^2$ be a nonempty nontrivial convex cone. Due to Luc (1985 and 1989), if $A$ is compact and if the closure $\overline C$ is pointed, then the efficient set $E(A|C)$ of $A$ w.r.t. $C$ is homeomorphic to a nonempty closed interval of $\R^1$, 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 $A\cap (\{a\}-\overline{C})$, for all $a\in A$. 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 $a\in A$ such that $A\cap (\{a\}-C)$ is bounded. Moreover, the weakly efficient set $E^w(A|C)$ w.r.t. any convex cone $C$ having nonempty interior is homeomorphic to a closed interval in $\mathbb R^1$ even if $C$ is not pointed.