On the theorem of Arrow-Barankin-Blackwell for weakly compact convex sets (Q1333931)

Let \(X\) be an ordered real vector space, with positive cone \(C\). Within a compact convex set \(A\), a point which is order-minimal is called efficient, while one minimizing a strictly positive functional is called positive proper efficient. Even in finite-dimensional spaces these are not equivalent; but the Arrow-Barankin-Blackwell theorem states that in \(\mathbb{R}^ n\), with \(C\) the nonnegative orthant, the set of positive proper efficient points of any compact convex body \(A\) is dense in the set of efficient points. This important theorem has previously been generalized to arbitrary real normed spaces, equipped with any ordering for which the positive cone has a base. It has also been generalized (by J. Jahn) to the case in which \(A\) is a weakly compact convex set. In this paper, the author derives a very strong generalization, for weakly compact convex sets in Hausdorff locally convex spaces, in which the positive cone is closed, convex, and has a bounded base.