A note on absolutely minimal extensions in finite metric spaces

Absolutely minimal Lipschitz extensions (AMLEs) are known to exist in many infinite metric settings, but the finite case is less settled. In metric spaces with at most four points, every function on a nonempty subset admits an AMLE in the sense that the Lipschitz constant cannot be further reduced on sets that are disjoint from the prescribed domain. We show that in five-point spaces such extensions may fail to exist.