ABB theorems: Results and limitations in infinite dimensions
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916329198452736 |
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| author | Daniilidis, Aris de Bernardi, Carlo Miglierina, Enrico |
| author_facet | Daniilidis, Aris de Bernardi, Carlo Miglierina, Enrico |
| contents | 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 |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10509 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | ABB theorems: Results and limitations in infinite dimensions Daniilidis, Aris de Bernardi, Carlo Miglierina, Enrico Functional Analysis Optimization and Control 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 |
| title | ABB theorems: Results and limitations in infinite dimensions |
| topic | Functional Analysis Optimization and Control |
| url | https://arxiv.org/abs/2407.10509 |