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

Given an Orlicz

-function

\varphi φ

and a positive decreasing weight

w 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} Λ φ , w

and

\lambda_{\varphi,w} λ φ , w

. We show that in the spaces

\Lambda_{\varphi,w} Λ φ , w

and

\lambda_{\varphi,w} λ φ , 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 φ

does not satisfy the appropriate

\Delta_2 Δ 2

-condition, while they have the Radon-Nikodým property if and only if

\varphi φ

satisfies the appropriate

\Delta_2 Δ 2

-condition.