Strictly convex norms and the local diameter two property

TL;DR

This paper investigates the relationship between strict convexity and the local diameter two property in Banach lattices, introducing a new monotonicity property and constructing specific norms with desired geometric features.

Contribution

It introduces a strict monotonicity property preventing the local diameter two property and constructs strictly convex norms on $c_0( ext{Gamma})$ with the local diameter two property.

Findings

Strict monotonicity property prevents local diameter two property.

Any strictly convex 1-symmetric norm on $c_0( ext{Gamma})$ has this property.

Existence of strictly convex norms on $c_0( ext{Gamma})$ with local diameter two property.

Abstract

We introduce and study a strict monotonicity property of the norm in solid Banach lattices of real functions that prevents such spaces from having the local diameter two property. Then we show that any strictly convex 1-symmetric norm on $c_0(\Gamma)$ possesses this property. In the opposite direction, we show that any Banach space which is strictly convex renormable and contains a complemented copy of $c_0(\mathbb N),$ admits an equivalent strictly convex norm for which the space has the local diameter two property. In particular, this enables us to construct a strictly convex norm on $c_0(\Gamma),$ where $\Gamma$ is uncountable, for which the space has a 1-unconditional basis and the local diameter two property.