Generalized Maxwell Formula for the Length of a Minimal Tree With a Given Topology / Ivanov A.O. and Tuzhilin A.A. // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2010. № 3. P. 7-14 [Moscow Univ. Math. Bulletin. Vol. 72, N 2, 2017. P. 100-106].
The classic Maxwell formula calculates the length of a planar locally minimal binary tree in terms of coordinates of its boundary vertices and directions of incoming edges. However, if an extreme tree with a given topology and a boundary has degenerate edges, then the classic Maxwell formula cannot be applied directly. In order to calculate the length of the extreme tree in this case, it is necessary to know which edges are degenerate. In this paper we generalize the Maxwell formula to arbitrary extreme trees in a Euclidean space of arbitrary dimension. Now, to calculate the length of such a tree, there is no need to know either what edges are degenerate, or the directions of nondegenerate boundary edges. The answer is the maximum of some special linear function on the corresponding compact convex subset of the Euclidean space coinciding with the intersection of some cylinders.
local minimal trees, Steiner problem, Maxwell formula.