Ball-covering property in uniformly non-\(l_3^{(1)}\) Banach spaces and application. (English) Zbl 1449.46018

Shang, Shaoqiang; Cui, Yunan

Abstr. Appl. Anal. 2013, Article ID 873943, 7 p. (2013).

Summary: This paper shows the following. (1) \(X\) is a uniformly non-\(l_3^{(1)}\) space if and only if there exist two constants \(\alpha, \beta > 0\) such that, for every 3-dimensional subspace \(Y\) of \(X\), there exists a ball-covering \(\mathfrak{B}\) of \(Y\) with \(c(\mathfrak{B}) = 4\) or \(5\) which is \(\alpha\)-off the origin and \(r(\mathfrak{B}) \leq \beta\). (2) If a separable space \(X\) has the Radon-Nikodým property, then \(X^*\) has the ball-covering property. Using this general result, we find sufficient conditions in order that an Orlicz function space has the ball-covering property.

MSC:

46B20

Geometry and structure of normed linear spaces

Keywords:

ball-covering property; uniformly non-\(l_3^{(1)}\) Banach spaces; Orlicz function space

References:

[1]	Cheng, L., Ball-covering property of Banach spaces, Israel Journal of Mathematics, 156, 111-123 (2006) · Zbl 1139.46016 · doi:10.1007/BF02773827
[2]	Cheng, L., Erratum to: ball-covering property of Banach spaces, Israel Journal of Mathematics, 184, 505-507 (2011) · Zbl 1257.46010 · doi:10.1007/s11856-011-0078-5
[3]	Cheng, L.; Cheng, Q.; Liu, X., Ball-covering property of Banach spaces that is not preserved under linear isomorphisms, Science in China A, 51, 1, 143-147 (2008) · Zbl 1152.46010 · doi:10.1007/s11425-007-0102-8
[4]	Cheng, L.; Shi, H.; Zhang, W., Every Banach space with a \(w\)-separable dual has a \(1 + \varepsilon \)-equivalent norm with the ball covering property, Science in China A, 52, 9, 1869-1874 (2009) · Zbl 1191.46010 · doi:10.1007/s11425-009-0175-7
[5]	Cheng, L.; Cheng, Q.; Shi, H., Minimal ball-coverings in Banach spaces and their application, Studia Mathematica, 192, 1, 15-27 (2009) · Zbl 1176.46015 · doi:10.4064/sm192-1-2
[6]	Cheng, L. X.; Luo, Z. H.; Liu, X. F.; Zhang, W., Several remarks on ball-coverings of normed spaces, Acta Mathematica Sinica, 26, 9, 1667-1672 (2010) · Zbl 1211.46006 · doi:10.1007/s10114-010-9036-0
[7]	Giesy, D. P., On a convexity condition in normed linear spaces, Transactions of the American Mathematical Society, 125, 114-146 (1966) · Zbl 0183.13204 · doi:10.1090/S0002-9947-1966-0205031-7
[8]	James, R. C., Uniformly non-square Banach spaces, Annals of Mathematics, 80, 542-550 (1964) · Zbl 0132.08902 · doi:10.2307/1970663
[9]	James, R. C., A nonreflexive Banach space that is uniformly nonoctahedral, Israel Journal of Mathematics, 18, 145-155 (1974) · Zbl 0292.46014 · doi:10.1007/BF02756869
[10]	Phelps, R. R., Convex Functions, Monotone Operators and Differentiability. Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics, 1364 (1989), New York, NY, USA: Springer, New York, NY, USA · Zbl 0658.46035 · doi:10.1007/BFb0089089
[11]	Fonf, V. P.; Zanco, C., Covering spheres of Banach spaces by balls, Mathematische Annalen, 344, 4, 939-945 (2009) · Zbl 1179.46015 · doi:10.1007/s00208-009-0336-6
[12]	Denker, M.; Hudzik, H., Uniformly non-\(l_n^{(1)}\) Musielak-Orlicz sequence spaces, Proceedings of the Indian Academy of Sciences, 101, 2, 71-86 (1991) · Zbl 0789.46008
[13]	Chen, S. T., Geometry of Orlicz spaces, Dissertationes Mathematicae, 356, 1-204 (1996) · Zbl 1089.46500