Intersection properties of the unit ball

Abstract

Let X be a real Banach space with the closed unit ball BX and the dual X⁎. We say that X has the intersection property(I) (general intersection property (GI), respectively) if, for each countable family (for each family, respectively) {Bi}i∈A of equivalent closed unit balls such that BX=⋂i∈ABi, one has BX⁎⁎=⋂i∈ABi∘∘, where Bi∘∘ is the bipolar set of Bi, that is, the bidual unit ball corresponding to Bi. In this paper we study relations between properties (I) and (GI), and geometric and differentiability properties of X. For example, it follows by our results that if X is Fréchet smooth or X is a polyhedral Banach space then X satisfies property (GI), and hence also property (I). Moreover, for separable spaces X, properties (I) and (GI) are equivalent and they imply that X has the ball generated property. However, properties (I) and (GI) are not equivalent in general. One of our main results concerns C(K) spaces: under a certain topological condition on K, satisfied for example by all zero-dimensional compact spaces and hence by all scattered compact spaces, we prove that C(K) satisfies (I) if and only if every nonempty Gδ-subset of K has nonempty interior.