Intersection properties of the unit ball

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) {B-i}(i is an element of A) of equivalent closed unit balls such that B-X = boolean AND(i is an element of A) B-i, one ha s B-X center dot center dot = boolean AND(i is an element of A) B-i(oo), where B-i(oo) is the bipolar set of B-i, that is, the bidual unit ball corresponding to 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 (CI) 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(delta)-subset of K has nonempty interior.