ABB theorems: Results and limitations in infinite dimensions

TL;DR

This paper constructs a specific example in infinite-dimensional space demonstrating the failure of a classical theorem, highlighting the importance of certain assumptions for the theorem's validity.

Contribution

It provides a counterexample in infinite dimensions showing the failure of the Arrow-Barankin-Blackwell theorem and clarifies the conditions needed for its validity.

Findings

Counterexample in infinite dimensions showing theorem failure

Maximal element not supported by any strictly positive functional

Bounded base assumption is crucial for theorem validity

Abstract

We construct a weakly compact convex subset of $\ell^2$ with nonempty interior that has an isolated maximal element, with respect to the lattice order $\ell _+^2$. Moreover, the maximal point cannot be supported by any strictly positive functional, showing that the Arrow-Barankin-Blackwell theorem fails. This example discloses the pertinence of the assumption that the cone has a bounded base for the validity of the result in infinite dimensions. Under this latter assumption, the equivalence of the notions of strict maximality and maximality is established