Intersection properties of the unit ball

Let $X$ be a real Banach space with the closed unit ball $B_X$ and the dual $X^{*}$. We say that $X$ has the {em intersection property} $I$ ({em general intersection property} $GI$, respectively) if, for each countable family (for each family, respectively) ${B_i}_{iin A}$ of equivalent closed unit balls such that $B_X=igcap_{iin A} B_i$, one has $B_{X^{**}}=igcap_{iin A} B_i^{circcirc}$, where $B_i^{circcirc}$ 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 Fr'echet 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 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.