Small combination of slices, dentability and stability results of small diameter properties in Banach spaces. (English) Zbl 1483.46006
The authors continue to study classical geometric properties of a Banach space \(X\), relative to the closed unit ball, \(B_X\). As is to be expected, these properties pass from \(X\) to its bidual (Proposition 2.1) and are preserved under \(\ell^p\) sums for \(1 \leq p < \infty \) (Proposition 2.8) and also in the case of Bochner integrable functions.
A study of the \(3\)-space problem for some of the properties is carried out in Section 4. Here again additional assumptions on \(Y\) or \(X/Y\) lead to positive answers. For example, if \(X/Y\) is strongly regular and \(Y\) has the BSCSP, then \(X\) has the BSCSP. Similarly, ball huskability behaves well when \(Y\) is of finite codimension.
A study of the \(3\)-space problem for some of the properties is carried out in Section 4. Here again additional assumptions on \(Y\) or \(X/Y\) lead to positive answers. For example, if \(X/Y\) is strongly regular and \(Y\) has the BSCSP, then \(X\) has the BSCSP. Similarly, ball huskability behaves well when \(Y\) is of finite codimension.
Reviewer: T.S.S.R.K. Rao (Bangalore)
MSC:
| 46B04 | Isometric theory of Banach spaces |
| 46B20 | Geometry and structure of normed linear spaces |
| 46B22 | Radon-Nikodým, Kreĭn-Milman and related properties |
References:
| [1] | Abrahamsen, T.; Lima, V.; Nygaard, O., Remarks on diameter 2 properties, J. Convex Anal., 20, 1, 439-452 (2013) · Zbl 1274.46027 |
| [2] | Acosta, M. D.; Becerra Guerrero, J.; López Pérez, G., Stability results of diameter 2 properties, J. Convex Anal., 22, 1, 1-17 (2015) · Zbl 1332.46010 |
| [3] | Basu, S., On ball dentable property in Banach spaces, (Math. Analysis and Its Applications in Modeling (ICMAAM 2018). Math. Analysis and Its Applications in Modeling (ICMAAM 2018), Springer Proceedings in Mathematics and Statistics, vol. 302 (2020)), 145-149 · Zbl 1455.46011 |
| [4] | Basu, S.; Rao, T. S.S. R.K., On small combination of slices in Banach spaces, Extr. Math., 31, 1-10 (2016) · Zbl 1367.46014 |
| [5] | Becerra Guerrero, J.; López Pérez, G.; Rueda Zoca, A., Octahedral norms and convex combination of slices in Banach spaces, J. Funct. Anal., 266, 4, 2424-2435 (2014) · Zbl 1297.46011 |
| [6] | Becerra Guerrero, J.; López Pérez, G.; Rueda Zoca, A., Big slices versus big relatively weakly open subsets in Banach spaces, J. Math. Anal. Appl., 428, 2, 855-865 (2015) · Zbl 1332.46011 |
| [7] | Becerra Guerrero, J.; López Pérez, G.; Rueda Zoca, A., Extreme differences between weakly open subsets and convex combination of slices in Banach spaces, Adv. Math., 269, 56-70 (2015) · Zbl 1312.46022 |
| [8] | Becerra Guerrero, J.; López Pérez, G.; Rueda Zoca, A., Subspaces of Banach spaces with big slices, Banach J. Math. Anal., 10, 771-782 (2016) · Zbl 1365.46007 |
| [9] | Bourgain, J., La propriété de Radon-Nikodym, Publ. Math. Univ. Pierre et Marie Curie, 36 (1979) · Zbl 0423.46011 |
| [10] | Bourgin, R. D., Geometric Aspects of Convex Sets with the Radon-Nikodym Property, Lecture Notes in Mathematics, vol. 993 (1983), Springer-Verlag: Springer-Verlag Berlin · Zbl 0512.46017 |
| [11] | Diestel, J.; Uhl, J. J., Vector Measures, Amer. Math. Soc., vol. 15 (1977) · Zbl 0369.46039 |
| [12] | Edgar, G. Ȧ; Wheeler, R., Topological properties of Banach spaces, Pac. J. Math., 115, 317-350 (1984) · Zbl 0506.46007 |
| [13] | Ghoussoub, N.; Maurey, B., \( G_\delta\) embeddings in Hilbert space, J. Funct. Anal., 61, 1, 72-97 (1985) · Zbl 0565.46011 |
| [14] | Ghoussoub, N.; Godefroy, G.; Maurey, B.; Scachermayer, W., Some Topological and Geometrical Structures in Banach Spaces, Mem. Amer. Math. Soc., vol. 70, 378 (1987) · Zbl 0651.46017 |
| [15] | Ghoussoub, N.; Maurey, B.; Schachermayer, W., Geometrical implications of certain infinite-dimensional decomposition, Trans. Am. Math. Soc., 317, 541-584 (1990) · Zbl 0717.46015 |
| [16] | Harmand, P.; Werner, D.; Werner, W., M-Ideals in Banach Spaces and Banach Algebras, Lecture Notes in Mathematics, vol. 1547 (1993), Springer-Verlag: Springer-Verlag Berlin · Zbl 0789.46011 |
| [17] | Langemets, J., Geometrical Structure in Diameter 2 Banach Spaces, Dissertationes Mathematicae Universitatis Tartuensis, vol. 99 (2015) · Zbl 1355.46003 |
| [18] | Langemets, J.; Pirk, K., Stability of diametral diameter two properties, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., 115 (2021) · Zbl 1506.46013 |
| [19] | Rosenthal, H. P., On the structure of non-dentable closed bounded convex sets, Adv. Math., 70, 1-58 (1988) · Zbl 0654.46024 |
| [20] | Schachermayer, W., The Radon Nikodym property and the Krein-Milman property are equivalent for strongly regular sets, Trans. Am. Math. Soc., 303, 2, 673-687 (1987) · Zbl 0633.46023 |
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.
