ABB Theorems: Results and Limitations in Infinite Dimensions

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, which shows 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.