Relationships between K-monotonicity and rotundity properties with application

TL;DR

This paper explores the relationships between k-rotundity, uniform K-monotonicity, reflexivity, and K-order continuity in symmetric spaces, providing characterizations and applications in approximation theory.

Contribution

It offers a complete characterization of decreasing uniform K-monotonicity and K-order continuity, and addresses the restriction of these properties to the positive cone of the space.

Findings

Characterization of decreasing uniform K-monotonicity in symmetric spaces

Complete description of K-order continuity in symmetric spaces

Application of geometric properties to approximation theory

Abstract

In this paper we investigate a relationship between fully k-rotundity properties, uniform K-monotonicity properties, reflexivity and K-order continuity in a symmetric spaces E. We also answer a crucial question whether fully k-rotundity properties might be restricted in definition to E^d the positive cone of all nonnegative and decreasing elements of E. We present a complete characterization of decreasing uniform K-monotonicity and K-order continuity in E. It is worth mentioning that we also establish several auxiliary results describing reflexivity in Lorentz spaces and K-order continuity in Orlicz spaces. Finally, we show an application of discussed geometric properties to the approximation theory.