Diameter of Weak Neighborhoods and the Radon-Nikodým Property in Orlicz-Lorentz Spaces

Given an Orlicz $N$-function $\varphi$ and a positive decreasing weight $w$, we present criteria for the diameter two property and for the Radon-Nikodým property in the Orlicz-Lorentz function and the sequence spaces $\Lambda_{\varphi,w}$ and $\lambda_{\varphi,w}$. We show that in the spaces $\Lambda_{\varphi,w}$ and $\lambda_{\varphi,w}$, equipped with the Luxemburg norm, the diameter of any relatively weakly open 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-Nikodým property if and only if $\varphi$ satisfies the appropriate $\Delta_2$-condition.