Enlargements of positive sets

TL;DR

This paper explores the concept of enlargements of positive sets in SSD spaces, establishing a framework that links these enlargements to representative functions and extends known results from Banach space theory.

Contribution

It introduces the notion of enlargements for positive sets in SSD spaces, characterizes extremal elements, and connects them to representative functions, extending previous monotone set results.

Findings

Characterization of smallest and largest enlargements in $ ext{E}(A)$

Existence of a bijection between closed enlargements and representative functions

Extension of monotone set results from Banach spaces to SSD spaces

Abstract

In this paper we introduce the notion of enlargement of a positive set in SSD spaces. To a maximally positive set $A$ we associate a family of enlargements $\E(A)$ and characterize the smallest and biggest element in this family with respect to the inclusion relation. We also emphasize the existence of a bijection between the subfamily of closed enlargements of $\E(A)$ and the family of so-called representative functions of $A$. We show that the extremal elements of the latter family are two functions recently introduced and studied by Stephen Simons. In this way we extend to SSD spaces some former results given for monotone and maximally monotone sets in Banach spaces.