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arXiv · 2603.03549

Order-Preserving Extensions of Hadamard Space-Valued Lipschitz Maps

Edoardo Gargiulo Efe A. Ok
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Abstract

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.

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Edoardo Gargiulo Efe A. Ok. 2026-03-03. Order-Preserving Extensions of Hadamard Space-Valued Lipschitz Maps. https://arxiv.org/abs/2603.03549

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