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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 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 $c_0(Γ)$ 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.

preprint2020arXivOpen access
Carlo Alberto De BernardiJacopo SomagliaLibor Vesely
math.FA
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Carlo Alberto De BernardiJacopo SomagliaLibor Vesely

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