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arXiv · 1902.04170

Bidual octahedral renormings and strong regularity in Banach spaces

Johann Langemets·Ginés López-Pérez
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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).

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Johann Langemets, Ginés López-Pérez. 2019-02-11. Bidual octahedral renormings and strong regularity in Banach spaces. https://doi.org/10.1017/s1474748019000264

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