Uniform K-monotonicity and K-order continuity in symmetric spaces with application to approximation theory

We investigate K-order continuity in a symmetric space E using the fundamental function ϕ of E. We also show a connection between reflexivity and K-order continuity in symmetric spaces. Next, we present several results devoted to a characterization of uniform K-monotonicity and decreasing (increasing) uniform K-monotonicity in symmetric spaces. We also discuss a relationship between decreasing (resp. increasing) uniform monotonicity and decreasing (resp. increasing) uniform K-monotonicity. Next, we deliberate a correlation between uniform K-monotonicity and uniform rotundity in symmetric spaces. Finally, employing K-monotonicity properties and K-order continuity we provide solvability and stability of the best approximation problem in the sense of the Hardy-Littlewood-Pólya relation ≺ in symmetric spaces.