Yunan Cui

Extreme Points and Strongly Extreme Points in Orlicz Spaces Equipped with the Orlicz Norm

Criteria for extreme points and strongly extreme points of the unit ball in Orlicz spaces with the Orlicz norm are given. These results are applied to characterization of extreme points of B(L +L∞) which corresponds with the result obtained by R. Grza̧ślewicz and H. Schaefer [10] and H. Schaefer [25]. Moreover, we show that every extreme point of B(L + L∞) is strongly extreme. We also get criteria for extreme points of B(L ∩ L∞) (1 ≤ p < ∞) using Theorem 1 and for strongly extreme points of B(Lp∩L∞) (1 ≤ p < ∞) applying Theorem 2. Although, criteria for extreme points of B(L ∩L∞) were known (see [13]), we can easily deduce them from our main results and we can extend those results to establish which among extreme points are strongly extreme. The descriptions of the extreme and strongly extreme points of B(L ∩ L∞) (1 < p < ∞) are original. Moreover, criteria for extreme points and strongly extreme points of the unit ball in the subspace of finite elements of an Orlicz space are deduced on the basis of our main results.

On the weakly strongly exposed property and some smoothness properties of Orlicz spaces

The notion of a weakly strongly exposed Banach space is introduced and it is shown that this property is the dual property of very smoothness. Criteria for this property in Orlicz function spaces equipped with the Orlicz norm are presented. Criteria for strong smoothness and very smoothness of their subspaces of order continuous elements in the case of the Luxemburg norm are also given.