Diameter two properties, convexity and smoothness

2016/06/01 by Abrahamsen, Trond A., Lima, Vegard, Nygaard, Olav +1
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1606.00221

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∈ 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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