Order-Preserving Extensions of Hadamard Space-Valued Lipschitz Maps

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.