On the density of proper efficient points

Abstract

In this paper, our aim is to discuss the density of proper efficient points. As an interesting application of the results in this paper, we want to prove a density theorem of Arrow, Barankin, and Blackwell. In [1], Luc introduced a new concept of the proper efficient point for a set. Using some results of recession cone, Luc established efficiency conditions, especially proper efficiency and domination properties ( [1, 2]). The present paper is devoted to the study of the density of proper efficient points. In detail, the set of proper efficient points for a set is dense in the set of efficient points. As an interesting application of the results in this paper, we prove a density theorem of Arrow, Barankin, and Blackwell ( [3,4]). First let us recall some notations: Throughout the paper, E is a separated locally convex topological linear space and E* its topological dual. U(O) denotes the family of balanced open convex neighbourhoods of the origin in E. For A c E, cone(A), cl(A), and int(A) denote the generated cone, the closure, and the interior of A, respectively. Let C C E be a convex cone, and let A be a nonempty subset of E. We say that x E A is an efficient point of A with respect to C if there exists y E A, such that y E x-C; then y E x + C. Equivalently, (x-C) n A c x + C. If the C is pointed (that is, C n (-C) =' o}), then x E A is an efficient point iff (x C) n A= {x}. We denote by E(A, C) the set of all efficient points of A (with respect to C). We say that x E A is a proper efficient point of A with respect to C if there exists a closed convex cone K =/ E such that C\{O} c int(K) and x E(A, K). The set of proper efficient points of A is denoted by Prop E(A, C). It is obvious that the set of proper efficient points of A is contained in the set of efficient points, Prop E(A, C) C E(A, C), but the converse is not generally true. If C is a convex cone, the convex set B c C is said to be a base of C if O0 cl(B) and C=cone(B)=U{tB:t>0}={tb:t>ObB}. A cone with base must be pointed. Received by the editors December 14, 1993 and, in revised form, October 3, 1994. 1991 Mathematics Subject Classification. Primary 90C31.