Functions with uniform level sets

TL;DR

This paper studies functions with uniform level sets on topological vector spaces, exploring their properties and applications in separation theorems, scalarization, and various areas like optimization and finance.

Contribution

It extends previous results on functions with uniform level sets, analyzing their properties and applications in separation and scalarization in topological vector spaces.

Findings

Functions with uniform level sets can be continuous, convex, or quasiconcave.

They can coincide with Minkowski functionals or order unit norms.

The core of a closed pointed convex cone equals its interior in an appropriate norm topology.

Abstract

Functions with uniform level sets can represent orders, preference relations or other binary relations and thus turn out to be a tool for scalarization that can be used, e.g., in multicriteria optimization, decision theory, mathematical finance, production theory and operator theory. Sets which are not necessarily convex can be separated by functions with uniform level sets. This has a deep impact on functional analysis, where many proofs require separation theorems. This report focuses on properties of real-valued and extended-real-valued functions with uniform level sets which are defined on a topological vector space. This includes the extension of aspects and results given in an earlier paper by Gerth (now Tammer) and Weidner. The functions may be, e.g., continuous, convex, strictly quasiconcave or sublinear. They can coincide with a Minkowski functional or with an order unit norm on a subset of the space. As a side result, we show that the core of a closed pointed convex cone is its interior in an appropriate norm topology.