On uniform convexity of Banach spaces. (English) Zbl 1231.46008
Summary: This paper gives some relations and properties of several kinds of generalized convexity in Banach spaces. As a result, it proves that every kind of uniform convexity implies the Banach-Saks property, and several notions of uniform convexity in the literature are actually equivalent.
MSC:
| 46B20 | Geometry and structure of normed linear spaces |
References:
| [1] | Clarkson, J.: Uniformly convex spaces. Trans. Amer. Math. Soc., 40, 396–414 (1936) · JFM 62.0460.04 · doi:10.1090/S0002-9947-1936-1501880-4 |
| [2] | Eshita, K., Takahashi, W.: On the uniform convexity of subsets of Banach spaces. Sci. Math. Jpn., 60, 577–594 (2004) · Zbl 1074.46009 |
| [3] | Johnson, W., Lindenstrauss, J.: Handbook of the Geometry of Banach Spaces I, North-Holland Publishing Co., Amsterdam, 2001 · Zbl 0970.46001 |
| [4] | Kutzarova, D.: {\(\kappa\)}-{\(\beta\)} and {\(\kappa\)}-nearly uniformly convex Banach spaces. J. Math. Anal. Appl., 162, 322–338 (1991) · Zbl 0757.46026 · doi:10.1016/0022-247X(91)90153-Q |
| [5] | Kutzarova, D., Rolewicz, S.: On the nearly uniformly convex set. Arch. Math. (Basel), 57, 385–394 (1991) · Zbl 0756.52004 |
| [6] | Lin, P. K.: {\(\kappa\)}-uniform rotundity is quivalent to {\(\kappa\)}-uniform convexity. J. Math. Anal. Appl., 132, 349–355 (1988) · Zbl 0649.46014 · doi:10.1016/0022-247X(88)90066-2 |
| [7] | Suyalatu: {\(\kappa\)}-super-strongly convex and {\(\kappa\)}-super-strongly smooth Banach spaces. J. Math. Anal. Appl., 1, 45–56 (2004) · Zbl 1094.46016 · doi:10.1016/j.jmaa.2004.02.020 |
| [8] | Zhao, M., Fang, B. X., Wang, Y. H.: {\(\kappa\)}-rotundities and their generalizations. Chinese Ann. Math. Ser. A, 21, 289–294 (2000) · Zbl 0960.46016 |
| [9] | Huff, R.: Banach spaces which are nearly uniformly convex. Rocky Mountain J. Math., 10, 743–749 (1980) · Zbl 0505.46011 · doi:10.1216/RMJ-1980-10-4-743 |
| [10] | Ishihara, H., Takahashi, W.: Modulus of convexity, character of convexity and fixed theorems. Kodai Math. J., 10, 197–208 (1987) · Zbl 0654.47041 · doi:10.2996/kmj/1138037414 |
| [11] | Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis I, American Mathematical Society, Providence, RI, 2000 · Zbl 0946.46002 |
| [12] | Diestel, J.: Geometry of Banach Spaces – Selected Topics 485, Springer-Verlag, New York, 1975 · Zbl 0307.46009 |
| [13] | Diestel, J.: Sequences and Series in Banach Spaces 92, Springer-Verlag, New York, 1984 · Zbl 0542.46007 |
| [14] | Cheng, L. X., Cheng, Q. J., Luo, Z. H., Zhang, W.: Every weakly compact set can be uniformly embedded into a reflexive Banach space. Acta Mathematica Sinica, English Series, 25(7), 1109–1112 (2009) · Zbl 1182.46004 · doi:10.1007/s10114-009-7545-5 |
| [15] | Cheng, L. X., Cheng, Q. J., Wang, B., Zhang, W.: On the super-weakly compact sets and uniformly convexifiable sets. Studia. Math., 199, 145–169 (2010) · Zbl 1252.46009 · doi:10.4064/sm199-2-2 |
| [16] | Cheng, L. X., Zhang, W.: A note on non-support points, negligible sets, Gateaux differentiability and Lipschitz embeddings. J. Math. Anal., 350(2), 531–536 (2009) · Zbl 1170.46039 · doi:10.1016/j.jmaa.2007.12.046 |
| [17] | Cheng, L. X., Teng, Y. M.: Certain subsets on which every bounded convex function is continuous. Acta Mathematica Sinica, English Series, 23(6), 1063–1066 (2007) · Zbl 1128.26008 · doi:10.1007/s10114-005-0835-7 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
