Basic theory of pp-Amemiya norm in Orlicz spaces (1≤p≤∞)(1≤p≤∞): Extreme points and rotundity in Orlicz spaces endowed with these norms

In any modular space generated by a convex modular we define a family of new norms (called pp-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 pp-Amemiya norm in Orlicz function spaces generated by NN-functions are discussed. The intervals for pp-Amemiya norm attainability are described. Criteria for extreme points as well as for rotundity of Orlicz function spaces endowed with pp-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 pp-Amemiya norm with 1