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

Let (E, ‖·‖E) be a symmetric space and let Y ⊂ X be a nonempty subset. For x ∈ X denote PY (x) = {y ∈ Y : ‖x− y‖ = dist(x, Y )}. Any element y ∈ PY (x) is called a best approximant in Y to x. A nonempty set Y ⊂ X is called proximinal or set of existence if PY (x) 6= ∅ for any x ∈ X. A nonempty set Y is said to be a Chebyshev set if it is proximinal and PY (x) is a singleton for any x ∈ E. A symmetric space E is said to be strictly K-monotone (shortly E ∈ (SKM)) if for any x, y ∈ E such that x∗ 6= y∗, x ≺ y we have ‖x‖E < ‖y‖E . A point x ∈ E is called a point of K-order continuity of E if for any (xn) ⊂ E such that xn ≺ x and xn → 0 a.e. we have ‖xn‖E → 0. A symmetric space E is called K-order continuous (shortly E ∈ (KOC)) if every element x of E is a point of K-order continuity. We present results devoted to application of strict K-monotonicity and K-order continuity in symmetric spaces. We characterize a relationship between strict Kmonotonicity, K-order continuity and the best dominated approximation problems with respect to the Hardy-Littlewood-Pólya relation ≺. First, using a local approach to strict K-monotonicity we show a necessary condition for uniqueness of the best dominated approximation under the relation ≺ in a symmetric space E. Next, we discuss a correlation between a point of K-order continuity and an existence of a best dominated approximant with respect to ≺. Finally, we present a full criteria, written in terms of K-order continuity, under which a closed and K-bounded above subset of a symmetric space E is proximinal. The above results come from the paper [1].