Strictly convex renormings and the diameter 2 property
Volume 274 / 2024
Studia Mathematica 274 (2024), 37-49
MSC: Primary 46B20; Secondary 46B22.
DOI: 10.4064/sm221216-28-8
Published online: 11 January 2024
Abstract
A Banach space (or its norm) is said to have the diameter $2$ property (D$2$P for 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-$1$ finite-codimensional projection it is impossible to be uniformly rotund in every direction and at the same time have the D$2$P.
