Functions with uniform level sets

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.