This is an announcement for the paper “Diameter two properties, convexity and smoothness” by Trond A. Abrahamsenhttp://arxiv.org/find/math/1/au:+Abrahamsen_T/0/1/0/all/0/1, Vegard Limahttp://arxiv.org/find/math/1/au:+Lima_V/0/1/0/all/0/1, Olav Nygaardhttp://arxiv.org/find/math/1/au:+Nygaard_O/0/1/0/all/0/1, Stanimir Troyanskihttp://arxiv.org/find/math/1/au:+Troyanski_S/0/1/0/all/0/1.
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..
The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1606.00221