Hardy-Littlewood-Pólya relation in the best dominated approximation in symmetric spaces

We investigate a correspondence between strict K-monotonicity, K-order continuity and the best dominated approximation problems with respect to the Hardy-Littlewood-Pólya relation ≺. Namely, we study, in terms of an LKM point and a UKM point, a necessary condition for uniqueness of the best dominated approximation under the relation ≺ in a symmetric space E. Next, we characterize a relation between a point of K-order continuity and an existence of a best dominated approximant with respect to ≺. Finally, we present a compete criteria, written in a notion of K-order continuity, under which a closed and K-bounded above subset of a symmetric space E is proximinal.