This is an announcement for the paper “Star-finite coverings of Banach spaces” by Carlo Alberto De Bernardihttps://arxiv.org/search/math?searchtype=author&query=De+Bernardi%2C+C+A, Jacopo Somagliahttps://arxiv.org/search/math?searchtype=author&query=Somaglia%2C+J, Libor Veselyhttps://arxiv.org/search/math?searchtype=author&query=Vesely%2C+L.

Abstract: 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 by 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 proving existence of a star-finite covering of $c_0(\Gamma)$ 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.

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/2002.04308