The Length of an Extremal Network in a Normed Space: Maxwell Formula

Abstract

In the present paper we consider local minimal and extremal networks in normed spaces. It is well known that in the case of the Euclidean space these two classes coincide and the length of a local minimal network can be found by using only the coordinates of boundary vertices and the directions of boundary edges (the Maxwell formula). Moreover, as was shown by Ivanov and Tuzhilin [15], the length of a local minimal network in the Euclidean space can be found by using the coordinates of boundary vertices and the structure of the network. In the case of an arbitrary norm there are local minimal networks that are not extremal networks, and an analogue of the formula mentioned above is only true for extremal networks; this is the main result of the paper. Moreover, we generalize the Maxwell formula for the case of extremal networks in normed spaces and give an explicit construction of norming functionals used in the formula for several normed spaces.