Extreme points and strongly extreme points in Orlicz spaces equipped with the Orlicz norm
The two main results of the present paper give a full description of the extreme points and the strongly extreme points of the unit ball of Orlicz spaces with the Orlicz norm, denoted by \( L_{ \Phi}^0 \) for convenient \( \Phi .\) The Amemiya formula for the Orlicz norm plays an important role not only in the proofs, which are very laborious, but in the statement of the two basic results. The authors use the result on extreme points for characterizing the spaces \( L_{ \Phi}^0 \) which are strictly convex and for getting some known results about extreme points in the unit balls of \( L_{ \infty},\) \( L_1 + L_{ \infty}\) and \( L_1 \cap L_{ \infty}.\) They characterize the strongly extreme points in the unit ball of \( L_1 \cap L_{ \infty}\) and prove that in \(L_{ \infty}\) and \( L_1 + L_{ \infty}\) every extreme point is strongly extreme. They also apply the main results to prove that \( L_p \cap L_{ \infty}\) \( ( 1<p< \infty )\) is strictly convex and to describe the strongly extreme points of its unit ball. The paper finishes with an application to the study of the space \( E_{ \Phi},\) the subspace of finite elements of \( L_{ \Phi}^0 ,\) where it is shown that every extreme point of the unit ball is an extreme point of the unit ball of \( L_{ \Phi}^0 \). From this it can be deduced that the set of strongly extreme points either is empty or coincides with the set of extreme points.
scientific article; zbMATH DE number 2075231