The Convex Point of Continuity Property in Asplund Spaces

Given a property of Banach spaces that is hereditary, it is natural to ask whether a Banach space has the property if each of its subspaces with a w Ž . Ž . particular structure such as a Schauder basis or a Schauder finite-dix mensional decomposition has the property. The motivation for such questions is that it is much easier to deal with Banach spaces with such an additional structure. A subset C of a Banach space has the convex point of continuity Ž . property, CPCP resp. point of continuity property, PCP , provided for Ž . each nonempty closed convex resp. nonempty closed subset D of C, the formal identity map I: D a D has a point of weak-to-norm continuity. A Ž . Banach space has the CPCP resp. PCP provided its closed unit ball has Ž . the CPCP resp. PCP . It is well known that the Radon]Nikodym property ́ Ž . RNP implies the PCP. Clearly, the PCP implies the CPCP. These three Ž w x. w x properties are indeed distinct cf. 4, 7 . See 5 for a splendid survey of these geometric properties.