Star-finite coverings of Banach spaces

We study star-finite coverings of infinite-dimensional normed spaces. A family of sets is called star-finite if each of its members intersects only finitely many other members of the family. It follows from our results that an LUR or a uniformly Fréchet smooth infinite-dimensional Banach space does not admit star-finite coverings by closed balls. On the other hand, we present a quite involved construction of a star-finite covering of c_0(Γ) by Fréchet smooth centrally symmetric bounded convex bodies. A similar but simpler construction shows that every normed space of countable dimension (and hence incomplete) has a star-finite covering by closed balls.