Two Generalizations of a Theorem of Arrow, Barankin, and Blackwell

Abstract

In 1953, Arrow, Barankin, and Blackwell proved that if $R^n $ is equipped with its natural ordering and if A is a closed convex subset of $R^n $, then the set of points in A that can be supported by strictly positive linear functionals is dense in the set of all efficient (maximal) points of A. In this note two generalizations of this result are given. The first of these is in the setting of a dual system and requires relatively weak assumptions on the ordering cone but a rather strong compactness assumption on the set A. The second generalization, which is in the setting of a locally convex space, relaxes the compactness assumption on the set A but demands more stringent assumptions on the ordering cone. This second result was recently obtained by Petschke for normed spaces [M. Petschke, “On a theorem of Arrow, Barankin and Blackwell”, SIAM J. Control Optim., 28 (1990), pp. 395–401]. The proof given here is substantially different from that given by Petschke.