KREIN-MILMAN-TYPE PROBLEMS FOR COMPACT MATRICIALLY CONVEX-SETS

Matricially convex sets are convex sets of matrices in which matrix-valued convex coefficients are admitted along with the usual scalar-valued convex coefficients. Accordingly, extremal elements of these sets are defined with respect to the larger class of coefficients. As in scalar-valued convexity theory, a Krein-Milman theorem is highly desirable. In this paper, Krein-Milman problems are formulated for compact matricially convex sets, and are solved in the special case where the matricially convex sets are generated by normals.