# Bidual octahedral renormings and strong regularity in Banach spaces

## TL;DR

This paper demonstrates that separable Banach spaces containing ℓ₁ can be renormed so their biduals are octahedral, and that dual spaces with separable preduals can be renormed to satisfy the strong diameter two property.

## Contribution

It answers Godefroy's 1989 question by showing the existence of such renormings for separable spaces and their duals, linking octahedrality and strong regularity.

## Key findings

- Separable Banach spaces with ℓ₁ can be renormed for octahedral biduals.
- Dual spaces with separable preduals can be renormed to have the strong diameter two property.
- Provides a positive answer to a longstanding question in Banach space theory.

## 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).