On local convexities of Orlicz spaces endowed with s-norms

Let Φ be an Orlicz function and LΦ(X,Σ,μ) be the corresponding Orlicz space on a non-atomic, σ-finite, complete measure space (X,Σ,μ). 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.