Strictly convex renormings and the diameter 2 property

TL;DR

This paper constructs an equivalent norm on L_1[0,1] that is weakly midpoint locally uniformly rotund and has the diameter 2 property, and proves limitations for Banach spaces with certain projections regarding uniform rotundity and the D2P.

Contribution

It introduces a new norm on L_1[0,1] with specific convexity and diameter properties, and establishes impossibility results for certain Banach spaces.

Findings

Constructed a weakly midpoint locally uniformly rotund norm with D2P on L_1[0,1]

Proved that Banach spaces with finite-co-dimensional projections cannot be both uniformly rotund in every direction and have D2P

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.