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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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