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 the classical convex point of continuity property and strong regularity are equivalent, but also when they are defined for an arbitrary locally convex topology containing the weak topology. We also show that while the unit ball $B$ of $c_0$ 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 enjoys strong regularity for $\tau$-open sets. Moreover, $B$ satisfies the diameter two property for the topology $\tau$, that is, every nonempty $\tau$-open subset of $B$ has diameter two even though every $\tau$-open subset of $B$ contains convex combinations of relative $\tau$-open subsets with arbitrarily small diameter, that is, $B$ fails the strong diameter two property for the topology $\tau$. This stresses the known extreme differences up to now between those diameter two properties from a topological point of view.