  We study the problem of extending an order-preserving real-valued Lipschitz
map defined on a subset of a partially ordered metric space without increasing
its Lipschitz constant and preserving its monotonicity. We show that a certain
type of relation between the metric and order of the space, which we call
radiality, is necessary and sufficient for such an extension to exist.
Radiality is automatically satisfied by the equality relation, so the classical
McShane-Whitney extension theorem is a special case of our main
characterization result. As applications, we obtain a similar generalization of
McShane's uniformly continuous extension theorem, along with some functional
representation results for radial partial orders.

Functional Analysis