Asymmetric norms, cones and partial orders
This interesting paper is devoted to the study of the relations among asymmetric norms and orders in linear spaces. Although some of the main examples of asymmetric norms are defined by means of orderings using the lattice structure of some classical Banach spaces, no systematic study of asymmetric linear lattices was done before. The author is able to show how this topic can be understood in a unified framework, introducing the main mathematical tools that allow to do this. It must be said that the main known applications of the theory of asymmetric normed spaces -- complexity analysis of algorithms, optimization --, use lattice-based asymmetric norms. The author defines three asymmetric norms relating cones, ordering and topology, and studies the relations between them. An order provides a cone, and the cone together with a norm provides asymmetric norms that are closely related to the normed lattice structure; the reciprocal relations are essentially true and are also analyzed. Some applications and nice examples are given. The last part of the paper is devoted to the ordering associated to what is called Yudin cone, as a part of a specific study of finite-dimensional spaces with lattice asymmetric norms.
scientific article; zbMATH DE number 6474003