Bidual octahedral renormings and strong regularity in Banach spaces

Abstract

We prove that every separable Banach space containing ${\ell }_1$ can be equivalently renormed so that its bidual space is octahedral, which answers, in the separable case, a question by Godefroy in 1989. As a direct consequence, we obtain that every dual Banach space, with a separable predual, failing to be strongly regular (that is, without convex combinations of slices with diameter arbitrarily small for some closed, convex and bounded subset) can be equivalently renormed with a dual norm to satisfy the strong diameter two property (that is, such that every convex combination of slices in its unit ball has diameter two).

Compared to the previous version, we have now fixed a mistake that appeared in Proposition 2.5 and added a new Corollary 4.3