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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
##### Keywords:
uniformly non-square; ball-covering; Banach space
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##### References:
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