Abstract. Let A R^{2}
be a nonempty closed convex subset and C R^{2
}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(AC) 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 ({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 E^{w}(AC) w.r.t. any convex cone C
having nonempty interior is homeomorphic to a closed interval in R^{1} even if C is not pointed.
