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Packing constant for Cesàro-Orlicz sequence spaces. (English) Zbl 1374.46024

Let \(X\) be a Banach space and \(B(X)\), \(S(X)\) denote the unit ball and the unit sphere of \(X\), respectively. A sequence of balls with centers at \(x_1, x_2, \dots\) in \(X\) and a fixed radius \(r>0\) is said to be packed into the unit ball if the following two conditions hold: (i) \(\|x_n\|\leq 1 -r\), \(n = 1, 2, \dots\); (ii) \(\|x_n-x_m\| > 2r\), \(n\not=m\), \(n,m=1,2,\dots\).
In the 1950’s, J. A. C. Burlak et al. [Proc. Glasg. Math. Assoc. 4, 22–25 (1958; Zbl 0087.10801)] gave the following formula of the packing constant for a Banach space \(X\): \[ P(X) =\sup\{r > 0:\;\exists\{x_n\}_{n=1}^\infty\subseteq B(X), \|x_n\|\leq 1-r,\;\|x_n-x_m\|>2r\;\text{ for}\;n \not=m\}. \] C. A. Kottman [Trans. Am. Math. Soc. 150, 565–576 (1970; Zbl 0208.37503)] has shown that \[ P(X)=\frac{D(X)}{2+D(X)}, \] where \(D(X)=\sup\{\text{sep}(\{x_n\}): \{x_n\}\subseteq S(X)\}\) and \(\text{sep}(\{x_n\})= \inf\{\|x_n-x_m\|:n\not= m\}\). He has also proved the following inequality for any infinite-dimensional Banach space \(X\): \[ \frac{1}{3}\leq P(X)\leq \frac{1}{2}. \]
The main purpose of the paper under review is calculating the Kottman constant and the packing constant of the Cesàro-Orlicz sequence spaces \((\text{ces})_\varphi\) defined by an Orlicz function \(\varphi\) equipped with the Luxemburg norm. In particular, the authors give two formulas in order to compute the constants. Moreover, they introduce a new constant \(\tilde{D}(X)\), which seems to be relevant to the packing constant.

MSC:

46B45 Banach sequence spaces
46B20 Geometry and structure of normed linear spaces
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References:

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