Diameter two properties, convexity and smoothness
Functional Analysis
2016-10-11 v2
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 : "For every slice of and every norm-one element in , there is a point in distance as close to 2 as we want." Spaces with this property are shown to have non-smooth bidual.
Cite
@article{arxiv.1606.00221,
title = {Diameter two properties, convexity and smoothness},
author = {Trond A. Abrahamsen and Vegard Lima and Olav Nygaard and Stanimir Troyanski},
journal= {arXiv preprint arXiv:1606.00221},
year = {2016}
}
Comments
Removed Proposition 2.7 from version [v1] because of a gap in the proof. arXiv admin note: text overlap with arXiv:1506.05237