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

TL;DR

This paper characterizes the extremal points of the unit ball in general Orlicz--Lorentz spaces, providing new insights into their structure and dual space functionals without restrictive assumptions.

Contribution

It offers the first comprehensive characterization of extreme, strongly extreme, and exposed points in broad Orlicz--Lorentz spaces, relaxing previous conditions.

Findings

Characterization of extreme points in Orlicz--Lorentz spaces

Necessary and sufficient conditions for dual space norm attainment

Characterization of supporting functionals

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.