Dually cone-boundedness of a set and applications

TL;DR

This paper introduces a generalized cone-boundedness concept in normed vector spaces, exploring its properties, relationships with existing notions, and applications to key results like conic cancellation and Rådström embedding.

Contribution

It defines a new weaker boundedness concept relative to cones and demonstrates its utility in extending fundamental results in cone analysis and vector space embeddings.

Findings

The new boundedness concept is weaker than existing notions.

It enables generalizations of conic cancellation rules.

It supports extensions of the Rådström embedding procedure.

Abstract

We introduce and study a generalized concept of boundedness of a subset of a normed vector space with respect to a cone, which is defined as lower boundedness of the images of the underlying set through all the positive functionals of the cone. We show that this is a weaker notion when compared to other similar ones and we explore several links with the existing literature. We subsequently demonstrate that this concept furnishes the properties required to obtain various generalizations of important results and techniques, including conic cancellation rules and the R{\aa}dstr\"om embedding procedure.