On a Theorem of Arrow, Barankin, and Blackwell

Abstract

A well-known theorem of Arrow, Barankin, and Blackwell states that if ${\bf R}^n $ is equipped with the natural ordering, then for every compact convex subset S of ${\bf R}^n $ the set of properly minimal elements of S is dense in the set of minimal elements of S. In this note a result of Jahn is used to show a generalization of the density theorem of Arrow, Barankin, and Blackwell. It will be shown that this theorem holds in a real normed space that is partially ordered by a convex cone with a closed bounded base.