Maximal Points of Convex Sets in Locally Convex Topological Vector Spaces: Generalization of the Arrow–Barankin–Blackwell Theorem

In 1953, Arrow, Barankin, and Blackwell proved that, if C is a nonempty compact convex set in Rn with its standard ordering, then the set of points in C maximizing strictly positive linear functionals is dense in the set of maximal points of C. In this paper, we present a generalization of this result. We show that that, if C is a compact convex set in a locally convex topological space X and if K is an ordering cone on X such that the quasi-interiors of K and the dual cone K* are nonempty, then the set of points in C maximizing strictly positive linear functionals is dense in the set of maximal points of C. For example, our work shows that, under the appropriate conditions, the density results hold in the spaces Rn, Lp(Ω, μ), 1≤p≤∞, lp, 1≤p≤∞, and C (Ω), Ω a compact Hausdorff space, when they are partially ordered with their natural ordering cones.