Bidual octahedral renormings and strong regularity in banach spaces

Abstract

We prove that every separable Banach space containing `1 can be equivalently renormed so that its bidual space is octahedral, which answers, in the separable case, a question in Godefroy (1989) [5]. 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).