Extreme points, strongly extreme points and exposed points in Orlicz--Lorentz spaces

Abstract

In this paper, we investigate the extremal structure of the unit ball in the most general classes of Orlicz--Lorentz spaces. the characterizations of extreme points, strongly extreme points, and exposed points are given for Orlicz--Lorentz function spaces $\Lambda_{\varphi,\omega}$ generated by an arbitrary Orlicz function $\varphi$ and a non--increasing weight function $\omega$, without assuming $\varphi$ is an $N$-function and $\omega$ is strict decreasing. Furthermore, we provide necessary and sufficient conditions for a functional in the dual space to attain its Luxemburg norm at $x \in \Lambda_{\varphi,\omega}$ without assuming that $\varphi$ is an $N$--function. The supporting functionals of $x \in \Lambda_{\varphi,\omega}$ are also characterized.