Modeling of optimal networks in Manhattan Geometry by means of linkages

In a broad sense, linkages are constructions made of rigid elements connected in such a way that some of their pairs can rotate around a common point. One of the main tasks related to the study of linkages is the description of possible hinge positions. Important results in this area are provided by the theorems of King [7], [8] and Kempe [2]. The main result of this paper is the constructive proof of the existence of a linkage that solves the optimization problem, namely the search for the shortest network connecting the boundary of $m \geqslant 1$ points in a space of dimension $n \geqslant 2$ with the Manhattan metric. This work is a continuation of the author's previous works [3], [4], which described mechanisms for constructing the shortest network in the Euclidean plane, as well as the minimal parametric network in Euclidean space of dimension $k \geqslant 2$.