The equivalence between CPCP and strong regularity under Krein- Milman property

Abstract

We obtain a result in the spirit of the well-known W. Schachermeyer and H. P. Rosenthal research about the equivalence between Radon-Nikodym and Krein-Milman properties, by showing that, for closed, bounded and convex subsets C of a separable Banach space, under Krein-Milman property for C, one has the equivalence between convex point of continuity property and strong regularity both defined for every locally convex topology on C, containing the weak topology on C. Then, under Krein-Milman property, not only convex point of continuity property and strong regularity are equivalent as defined for weak topology, but even when they are defined for a locally convex topology containing the weak topology. We also show that while the unit ball B of c0 fails convex point of continuity property and strong regularity (both defined for the weak topology), threre is a locally convex topology tau on B, containing the weak topology on B, such that B still fails convex point of continuity property for tau , but B surprisingly satisfies strong regularity for tau-open subsets. As a consequence, using the usual norm of c0, we obtain that B satisfies the diameter two property for the topology tau , that is, every nonempty tau-open subset of B has diameter two, but every tau-open subset of B contains convex combinations of relative tau-open subsets with diameter arbitrarily small, that is, B fails strong diameter two property for topology tau , which stresses the known extreme difeerences up to now between those diameter two properties from a topological point of view.