Geometry of Orlicz spaces equipped with norms generated by some lattice norms in $\mathbb{R}^{2}$

TL;DR

This paper explores the geometric properties of Orlicz spaces with norms derived from lattice norms in , analyzing monotonicity, convexity, and isometric embeddings based on the generating functions and norms.

Contribution

It introduces a family of norms in Orlicz spaces based on lattice norms in  and characterizes their geometric properties and isometric structures.

Findings

Criteria for strict monotonicity and uniform monotonicities are established.

Conditions for strict convexity and isometric embeddings are provided.

Analysis includes subspaces of order continuous elements and their geometric features.

Abstract

In Orlicz spaces generated by convex Orlicz functions a family of norms generated by some lattice norms in $\mathbb{R}^{2}$ are defined and studied. This family of norms includes the family of the p-Amemiya norms ($1\leq p\leq\infty$) studied in [10-11], [14-15] and [20]. Criteria for strict monotonicity, lower and upper local uniform monotonicities and uniform monotonicities of Orlicz spaces and their subspaces of order continuous elements, equipped with these norms, are given in terms of the generating Orlicz functions, and the lattice norm in $\mathbb{R}^{2}$. The problems of strict convexity and of the existence of order almost isometric as well as of order isometric copies in these spaces are also discussed.