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Several remarks on ball-coverings of normed spaces. (English) Zbl 1211.46006
A ball-covering \(\mathcal B\) of a Banach space is a collection of open balls off the origin whose union contains the unit sphere of the space. A ball-covering of \(X\) is said to be minimal provided that its cardinality \(\mathcal B^\sharp_{\min}\) is the smallest among all cardinalities of ball-coverings of \(X\). If \(\mathcal B = \{B(x_j, r_j)\}\) is a ball-covering of \(X\), then \(r(\mathcal B) = \sup_j r_j\) is called the radius of \(\mathcal B\) and \(\mathcal B\) is said to be \(\alpha\)-off the origin if \(\|x_j\| - r_j \geq \alpha\) for all \(j\).
This paper presents two examples related to results of L.-X. Cheng [Isr. J. Math. 156, 111–123 (2006; Zbl 1139.46016)] and L.-X. Cheng, Q.-J. Cheng, H.-H. Shi [Stud. Math. 192, No. 1, 15–27 (2009; Zbl 1176.46015)]. The first one shows that for \(X = \ell_1^{(n)}\), \(n \geq 3\), although \(X\) contains an isometric copy of \(\ell_\infty^{(2)}\), we have \(\mathcal B^\sharp_{\min}= n+ 1\). The second one presents a four-dimensional space \(X\) with \(\mathcal B^\sharp_{\min} = 6\) that does not split in an \(\ell_\infty\)-sum of two nontrivial subspaces.
The following theorem is proved: \(X\) is uniformly non-square if and only if there exist \(\alpha, \beta > 0\) such that for every two-dimensional subspace \(Y \subset X\) there is a ball-covering \(\mathcal B\) of \(Y\) consisting of three balls such that \(r(\mathcal B) \leq \beta\) and \(\mathcal B\) is \(\alpha\) off the origin.

46B04 Isometric theory of Banach spaces
46B07 Local theory of Banach spaces
46B20 Geometry and structure of normed linear spaces
Full Text: DOI
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