Let be a normed space, be a proper closed convex cone with nonempty interior. Define the positive polar cone by . Let be a weak*-compact convex base of , i.e, is weak*-compact convex, and for any nonzero there exists unique and such that .
Given , , and define . By the weak*-compactness of , the infimum is attained.
Question: Show that it is attained at an extreme point of , i.e. is an extreme point if there does not exist two distinct , and such that .
It seems this can be done by applying the Krein Milman theorem.