Exposed points of matrix convex sets

 

Matrix convex sets extend the classical notion of convexity to the noncommutative setting, where sets are closed under convex combinations with matrix-valued coefficients. In classical convexity, an exposed point can be weakly separated from the set, a concept central to convexity. 

In this talk we investigate an analogous notion for matrix convex sets, defining and studying matrix exposed points. We establish a connection between matrix exposed and matrix extreme points: a matrix extreme point is exposed in the classical sense if and only if it is matrix exposed. This result leads to a Krein-Milman type spanning theorem for matrix exposed points, akin to the classical results of Straszewicz and Klee. Specifically, any compact matrix convex set can be generated by its matrix exposed points via (limits of) matrix convex combinations. Moreover, with similar techniques, an even stronger result is obtained, namely that the matrix exposed points are dense in the matrix extreme points.