Abstract
We study Banach spaces with the property that, given a finite number of slices of the unit ball, there exists a direction such that all these slices contain a line segment of length almost 2 in this direction. This property was recently named the symmetric strong diameter two property by Abrahamsen, Nygaard, and Põldvere. The symmetric strong diameter two property is not just formally stronger than the strong diameter two property (finite convex combinations of slices have diameter 2). We show that the symmetric strong diameter two property is only preserved by \(\ell _\infty \)-sums, and working with weak star slices we show that \(\text {Lip}_0(M)\) have the weak star version of the property for several classes of metric spaces M.
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Acknowledgements
The authors wish to express their thanks to Indrek Zolk for his collaboration in proving Lemma 5.4
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R. Haller, J. Langemets, and R. Nadel were partially supported by institutional research funding IUT20-57 of the Estonian Ministry of Education and Research.
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Haller, R., Langemets, J., Lima, V. et al. Symmetric Strong Diameter Two Property. Mediterr. J. Math. 16, 35 (2019). https://doi.org/10.1007/s00009-019-1306-1
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DOI: https://doi.org/10.1007/s00009-019-1306-1