Extreme points, strongly extreme points and exposed points in Orlicz--Lorentz spaces
Di. Wang, Yongjin. Li
TL;DR
This work investigates the extremal geometry of the unit ball in general Orlicz–Lorentz spaces \(\Lambda_{\varphi,\omega}\) by characterizing extreme, strongly extreme, and exposed points under minimal assumptions: arbitrary \(\varphi\) (not necessarily an \(N\)-function) and a nonincreasing weight \(\omega\) (not necessarily strictly decreasing). Employing a framework built on measure-preserving rearrangements, level-function techniques, and Köthe duality with \(\mathcal{M}_{\psi,\omega}\), the authors derive necessary and sufficient conditions for norm-attainment of dual functionals, identify supporting functionals, and give explicit criteria for exposed points. A key result is that strong extremity coincides with the \(\Delta_{2}\) condition on \(\varphi\) together with extremity, while extreme and exposed points are described via modular constraints \(\rho_{\varphi,\omega}(x)\), decompositions, and level-set measures. These findings extend prior work by relaxing strict decrease and \(N\)-function assumptions, enriching the understanding of the geometry and duality of Orlicz–Lorentz spaces with potential implications for related functional-analytic and geometric applications.
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 $Λ_{\varphi,ω}$ generated by an arbitrary Orlicz function $\varphi$ and a non--increasing weight function $ω$, without assuming $\varphi$ is an $N$-function and $ω$ is strict decreasing. Furthermore, we provide necessary and sufficient conditions for a functional in the dual space to attain its Luxemburg norm at $x \in Λ_{\varphi,ω}$ without assuming that $\varphi$ is an $N$--function. The supporting functionals of $x \in Λ_{\varphi,ω}$ are also characterized.