Representations of cones and applications to decision theory
Year: 2025
Type: article
Abstract: Let C be a cone in a locally convex Hausdorff topological vector space X containing 0. We show that there exists a (essentially unique) nonempty family K of nonempty subsets of the topological dual X ′ such that C = { x ∈ X : ∀ K ∈ K , ∃ f ∈ K , f ( x ) ≥ 0 } . Then, we identify the additional properties on the family K which characterize, among others, closed convex cones, open convex cones, closed cones, and convex cones. For instance, if X is a Banach space, then C is a closed cone if and only if the family K can be chosen with nonempty convex compact sets. These representations provide abstract versions of several recent results in decision theory and give us the proper framework to obtain new ones. This allows us to characterize preorders which satisfy the independence axiom over certain probability measures, answering an open question in Hara et al. (2019) [20] .
Authors: Paolo Leonetti, Giulio Principi
Institutions: University of Insubria, New York University
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
Funder: New York University
OpenAlex ID: W4409247222