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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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