This is an announcement for the paper "Big slices versus big relatively weakly open subsets in Banach spaces" by Julio Becerra Guerrero, Gines Lopez Perez and Abraham Rueda Zoido.

Abstract: We study the unknown differences between the size of slices and relatively weakly open subsets of the unit ball in Banach spaces. We show that every Banach space containing isomorphic copies of $c_0$ can be equivalently renormed so that every slice of its unit ball has diameter 2 and satisfying that its unit ball contains nonempty relatively weakly open subsets with diameter strictly less than 2, which answers by the negative an open problem. As a consequence a Banach space is constructed satisfying that every slice of its unit ball has diameter 2 and containing nonempty relatively weakly open subsets of its unit ball with diameter arbitrarily small, which stresses the differences between the size of slices and relatively weakly open subsets of the unit ball of Banach spaces.

Archive classification: math.FA

Mathematics Subject Classification: 46B20, 46B22

Remarks: 12 pages

Submitted from: glopezp@ugr.es

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

http://front.math.ucdavis.edu/1304.4397

or

http://arXiv.org/abs/1304.4397