Generalized Maxwell formula for the length of a minimal tree with a given topologyстатья
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Аннотация: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, 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.