Order-Preserving Extensions of Hadamard Space-Valued Lipschitz Maps

TL;DR

This paper investigates the extension of order-preserving Lipschitz functions from subsets of Hilbert spaces into Hadamard posets, revealing limitations in higher dimensions and establishing that no general order-theoretic Kirszbraun's theorem exists.

Contribution

It characterizes when such Lipschitz extensions are possible, showing they are only feasible in one-dimensional cases or trivial orders in higher dimensions.

Findings

Extensions are always possible in 1D cases.

In higher dimensions, extensions exist only for trivial orders.

No general order-theoretic Kirszbraun's theorem exists in higher dimensions.

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.