This is an announcement for the paper "Almost square Banach spaces" by Trond A. Abrahamsen, Johann Langemets, and Vegard Lima.

Abstract: We single out and study a natural class of Banach spaces -- almost square Banach spaces. These spaces have duals that are octahedral and finite convex combinations of slices of the unit ball of an almost square space have diameter 2. We provide several examples and characterizations of almost square spaces. In an almost square space we can find, given a finite set $x_1,x_2,\ldots,x_N$ in the unit sphere, a unit vector $y$ such that $|x_i+y|$ is almost one. We prove that non-reflexive spaces which are M-ideals in their biduals are almost square. We show that every space containing a copy of $c_0$ can be renormed to be almost square. A local and a weak version of almost square spaces are also studied.

Archive classification: math.FA

Mathematics Subject Classification: 46B20, 46B04, 46B07

Remarks: 22 pages

Submitted from: veli@hials.no

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1402.0818

or

http://arXiv.org/abs/1402.0818