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arXiv · 2407.10509

ABB theorems: Results and limitations in infinite dimensions

Aris Daniilidis·Carlo de Bernardi·Enrico Miglierina
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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

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Aris Daniilidis, Carlo de Bernardi, Enrico Miglierina. 2024-07-15. ABB theorems: Results and limitations in infinite dimensions. https://arxiv.org/abs/2407.10509

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