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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.

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