Theoretical location of optima via analysis techniques

In Hausdorff locally convex topological vector spaces, the Krein-Milman theorem allow us to describe compact convex sets as the closed convex hull of their extreme points. This coupled with measure-theoretic inequalities from the theory of probability, gives us a theoretical proof of where the optima of certain programs can be located. We aim to extend this results to abstract convexities in more general settings.