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

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 ${\mathrm{\Lambda }}_{\phi ,\omega }$ generated by an arbitrary Orlicz function $\phi$ and a non–increasing weight function $\omega$, without assuming $\phi$ 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 $x\in {\mathrm{\Lambda }}_{\phi ,\omega }$ without assuming that $\phi$ is an $N$–function. The supporting functionals of $x\in {\mathrm{\Lambda }}_{\phi ,\omega }$ are also characterized.