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Minimal ball-coverings in Banach spaces and their application. (English) Zbl 1176.46015
A ball-covering $${\mathcal B}$$ of a Banach space $$X$$ a is collection of open balls off the origin in $$X$$ and whose union contains the unit sphere of $$X$$. A ball-covering $${\mathcal B}$$ is called minimal if its cardinality is the smallest one among all ball-coverings of $$X$$. It is shown that, for every $$n, k \in {\mathbb N}$$ with $$k\leq n$$, there exists an $$n$$-dimensional space (namely, the $$\ell_\infty$$-sum of $$\ell_\infty^{(k-1)}$$ and $$\ell_2^{(n-k+1)}$$) admitting a minimal ball-covering of $$n+k$$ balls. Characterizations of uniformly non-square and superreflexive spaces in terms of ball-coverings are presented. Finally, it is shown that every infinite-dimensional Banach space admits an equivalent norm such that there is an infinite-dimensional quotient space (not necessarily separable) possessing a countable ball-covering.

##### MSC:
 46B03 Isomorphic theory (including renorming) of Banach spaces
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