Diameter of weak neighborhoods and the Radon-Nikodym property in Orlicz-Lorentz spaces

TL;DR

This paper characterizes when Orlicz-Lorentz spaces have the diameter two property and the Radon-Nikodym property based on the $ riangle_2$ condition of the Orlicz function, linking geometric and measure-theoretic properties.

Contribution

It provides new criteria linking the $ riangle_2$ condition of the Orlicz function to the diameter two and Radon-Nikodym properties in Orlicz-Lorentz spaces.

Findings

Spaces have diameter two property iff $ riangle_2$ condition fails.

Spaces have Radon-Nikodym property iff $ riangle_2$ condition holds.

Characterizes geometric properties of Orlicz-Lorentz spaces.

Abstract

Given an Orlicz $N$-function $\varphi$ and a positive decreasing weight $w$, we present criteria of the diameter two property and of the Radon-Nikod\'ym property in Orlicz-Lorentz function and sequence spaces $\Lambda_{\varphi,w}$ and $\lambda_{\varphi,w}$. We show that in the spaces $\Lambda_{\varphi,w}$ or $\lambda_{\varphi,w}$ equipped with the Luxemburg norm, the diameter of any relatively weakly subset of the unit ball in these spaces is two if and only if $\varphi$ does not satisfy the appropriate $\Delta_2$ condition, while they have the Radon-Nikod\'ym property if and only if $\varphi$ satisfies the appropriate $\Delta_2$ condition.