Basic theory of [formula omitted]-Amemiya norm in Orlicz spaces [formula omitted]: Extreme points and rotundity in Orlicz spaces endowed with these norms

Abstract

In any modular space generated by a convex modular we define a family of new norms (called p -Amemiya norms) which are equivalent to the Orlicz norm as well as to the Luxemburg norm. Next, the new case of Orlicz spaces is studied carefully. The attainable points of the p -Amemiya norm in Orlicz function spaces generated by N -functions are discussed. The intervals for p -Amemiya norm attainability are described. Criteria for extreme points as well as for rotundity of Orlicz function spaces endowed with p -Amemiya norm are given. The obtained results unify, complete and extend as well the results presented by a number of papers devoted to studying the geometry of Orlicz spaces endowed with the Luxemburg norm and the Orlicz norm separately. From our results it follows that there are Orlicz spaces which are rotund for p -Amemiya norm with 1 < p < ∞ only, that is, they are neither rotund for the Luxemburg norm corresponding to the case p = ∞ nor for Orlicz norm corresponding to the case p = 1 .