On local convexities of Orlicz spaces endowed with $s$-norms
Colloquium Mathematicum
MSC: Primary 46E30; Secondary 46B20
DOI: 10.4064/cm9702-10-2025
Published online: 1 April 2026
Abstract
Let $\varPhi $ be an Orlicz function and $L^\varPhi (X, \varSigma , \mu )$ be the corresponding Orlicz space on a non-atomic, $\sigma $-finite, complete measure space $(X,\varSigma ,\mu )$. We describe the local uniform convexity of Orlicz spaces endowed with the $s$-norm and discuss the weak and compact variants of this property. Also, we derive some results in approximation theory, concerning best approximations and farthest points. Thus, our study provides a comprehensive generalization of several results that have been obtained for Orlicz spaces with the Orlicz norm and the Luxemburg norm.
