Rotundity in transitive and separable Banach spaces

Abstract

In every separable Banach space the set of smooth points of the unit ball is a Gδ dense subset of the unit sphere (see [12]). In this paper, we find some conditions in order to obtain a similar result for rotund points. For instance, we prove that if the unit ball of a smooth and separable Banach space is free of rotund points then the set of non-norm-attaining functionals of norm 1 is residual in the unit sphere of the dual. Furthermore, by taking profit of these techniques we provide positive approaches to the Banach-Mazur conjecture for rotations.

Keywords: Rotund, smooth, Gδ-dense, transitive, separable

Quaestiones Mathematicae 30(2007), 85–96