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arXiv · 2212.13606

Strictly convex renormings and the diameter 2 property

Olav Nygaard·Märt Põldvere·Stanimir Troyansky·Tauri Viil
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Abstract

A Banach space (or its norm) is said to have the diameter $2$ property (D$2$P in short) if every nonempty relatively weakly open subset of its closed unit ball has diameter $2$. We construct an equivalent norm on $L_1[0,1]$ which is weakly midpoint locally uniformly rotund and has the D$2$P. We also prove that for Banach spaces admitting a norm-one finite-co-dimensional projection it is impossible to be uniformly rotund in every direction and at the same time have the D$2$P.

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Olav Nygaard, Märt Põldvere, Stanimir Troyansky, Tauri Viil. 2022-12-27. Strictly convex renormings and the diameter 2 property. https://arxiv.org/abs/2212.13606

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