Steiner points in $l_\infty^2$ spaсe

It is proved that for a given set of pairwise distinct points $x_1, \dots, x_n$ the sum of the distances from these points to their Steiner point in $l_\infty^2$ space is equal to the maximum of the sum of lengths of $[\frac{n}{2}] - 1$ separate segments and either a semi-perimeter of a triangle, or another segment with vertices in this set. The case of coincident points among $x_1, \dots, x_n$ is also studied.