Planar Manhattan local minimal and critical networks

The authors consider a version of the minimum Steiner tree problem for the l¹-induced distance function. A network connecting a given set of points in l¹ space is called locally minimal if no auxiliary vertex of the network can be moved to decrease the total length. It is called critical if there is no infinitesimal perturbation of the network which would decrease the total length. The authors show that unlike in the case of the l²-induced distance, a locally minimal network does not have to be critical. They also demonstrate some properties of critical and locally minimal networks.