Abstract
We characterise the class of those Banach spaces in which every convex combination of slices of the unit ball intersects the unit sphere as the class of those spaces in which every convex combination of slices of the unit ball contains two points at distance exactly two. Also, we study when the convex combinations of slices of the unit ball are relatively open or have non-empty relative interior for different topologies, studying the relationship between them and studying these properties for \(L_{\infty }\)-spaces and preduals of \(L_1\)-spaces.
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Abrahamsen, T.A., Becerra Guerrero, J., Haller, R., Lima, V., Põldvere, M.: Banach Spaces Where Convex Combinations of Relatively Weakly Open Subsets of the Unit Ball are Relatively Weakly Open, to appear in Studia Math. https://doi.org/10.4064/sm181016-10-1
Abrahamsen, T.A., Lima, V.: Relatively weakly open convex combination of slices. Proc. Am. Math. Soc. 146, 4421–4427 (2018)
Abrahamsen, T.A., Lima, V., Nygaard, O.: Remarks on diameter 2 properties. J. Conv. Anal. 20(2), 439–452 (2013)
Becerra Guerrero, J., López-Pérez, G., Rueda Zoca, A.: Octahedral norms in spaces of operators. J. Math. Anal. Appl. 427, 171–184 (2015)
Becerra Guerrero, J., López-Pérez, G., Rueda Zoca, A.: Subspaces of Banach spaces with big slices. Banach J. Math. Anal. 10, 771–782 (2016)
Ghoussoub, N., Godefroy, G., Maurey, B., Schachermayer, W.: Some topological and geometrical structures in Banach spaces. Mem. Am. Math. Soc. 378 (1987)
Haller, R., Kuuseok, P., Põldvere, M.: On Convex Combination of Slices of the Unit Ball in Banach Spaces, Preprint 2017. ar**v:1703.02919
Harmand, P., Werner, D., Werner, W.: \(M\)-Ideals in Banach Spaces and Banach Algebras. Lecture Notes in Mathematics, vol. 1547. Springer, Springer (1993)
Langemets, J., Lima, V., Rueda Zoca, A.: Octahedral norms in tensor products of Banach spaces. Q. J. Math. 68, 1247–1260 (2017)
Ryan, R.A.: Introduction to Tensor Products of Banach Spaces, Springer Monographs in Mathematics. Springer, London (2002)
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The authors are grateful to the anonymous referee for the valuable suggestions which have improved the exposition of the paper.
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The research of Ginés López-Pérez and Miguel Martín has been partially supported by Spanish MINECO/FEDER Grant number MTM2015-65020-P, by PCG2018-093794-B-I00 (MCIU/AEI/FEDER, UE), by (FEDER) Junta de Andalucía Grant A-FQM-484-UGR18 and Junta de Andalucía/FEDER Grant FQM-185. The research of Abraham Rueda Zoca has been supported by MECD (Spain) FPU2016/00015, Spanish MINECO/FEDER Grant number MTM2015-65020-P, by PCG2018-093794-B-I00 (MCIU/AEI/FEDER, UE), by (FEDER) Junta de Andalucía Grant A-FQM-484-UGR18 and Junta de Andalucía/FEDER Grant FQM-185.
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López-Pérez, G., Martín, M. & Rueda Zoca, A. Strong Diameter Two Property and Convex Combinations of Slices Reaching the Unit Sphere. Mediterr. J. Math. 16, 122 (2019). https://doi.org/10.1007/s00009-019-1403-1
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DOI: https://doi.org/10.1007/s00009-019-1403-1