Intersection properties of the unit ball

Abstract

Abstract Let X be a real Banach space with the closed unit ball B X 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) { B i } i ∈ A of equivalent closed unit balls such that B X = ⋂ i ∈ A B i , one has B X ⁎ ⁎ = ⋂ i ∈ A B i ∘ ∘ , where B i ∘ ∘ is the bipolar set of B i , that is, the bidual unit ball corresponding to B i . 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 Frechet 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 G δ -subset of K has nonempty interior.