Order-Preserving Extensions of Hadamard Space-Valued Lipschitz Maps
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917312174489600 |
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| author | Ok, Edoardo Gargiulo Efe A. |
| author_facet | Ok, Edoardo Gargiulo Efe A. |
| contents | We study the problem of extending any order-preserving Lipschitz function that maps a subset of a partially ordered Hilbert space X into a Hadamard poset Y without increasing its Lipschitz constant and preserving its monotonicity. This sort of an extension is always possible when X is one-dimensional. However, when dim X is at least 2 and Y satisfies some fairly weak conditions, it holds (universally) if and only if the order of X is trivial. The conditions on Y are satisfied by any Hilbert poset. Therefore, as a special case of our main result, we find that there is no order-theoretic generalization of Kirszbraun's theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_03549 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Order-Preserving Extensions of Hadamard Space-Valued Lipschitz Maps Ok, Edoardo Gargiulo Efe A. Functional Analysis We study the problem of extending any order-preserving Lipschitz function that maps a subset of a partially ordered Hilbert space X into a Hadamard poset Y without increasing its Lipschitz constant and preserving its monotonicity. This sort of an extension is always possible when X is one-dimensional. However, when dim X is at least 2 and Y satisfies some fairly weak conditions, it holds (universally) if and only if the order of X is trivial. The conditions on Y are satisfied by any Hilbert poset. Therefore, as a special case of our main result, we find that there is no order-theoretic generalization of Kirszbraun's theorem. |
| title | Order-Preserving Extensions of Hadamard Space-Valued Lipschitz Maps |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2603.03549 |