Abstract
We study smoothness and strict convexity of (the bidual) of Banach spaces in the presence of diameter 2 properties.We prove that the strong diameter 2 property prevents the bidual from being strictly convex and being smooth, and we initiate the investigation whether the same is true for the (local) diameter 2 property. We also give characterizations of the following property for a Banach space \({X}\): “For every slice \({S}\) of \({B_X}\) and every norm-one element \({x}\) in \({S}\), there is a point \({y \in S}\) in distance as close to 2 as we want.” Spaces with this property are shown to have non-smooth bidual.
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The main results of this paper were presented in a talk by the fourth named author at the Seminario Matematico e Fisico di Milano in March 2016. The fourth named author was partially supported by MTM2014-54182-P and the Bulgarian National Scientific Fund under Grant DFNI-I02/10.
Lecture given by S. Troyanski at the Seminario Matematico e Fisico di Milano on March 14, 2016
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Abrahamsen, T.A., Lima, V., Nygaard, O. et al. Diameter Two Properties, Convexity and Smoothness. Milan J. Math. 84, 231–242 (2016). https://doi.org/10.1007/s00032-016-0258-1
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DOI: https://doi.org/10.1007/s00032-016-0258-1