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 property (D2P 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 D2P. 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 D2P.
