The Degrees of the Vertices in Shortest Trees

Abstract

This chapter is addressed to the degrees of the vertices in shortest trees, since the properties of the degrees are important structural properties of the networks. On the one hand, the determination of these numbers is the problem to classify singularities in network minimizing for arbitrary norms. On the other hand, this question is closely related to the theory of singularities in energy-minimizing surfaces. The latter task is of great practical relevance: Soap films, grain boundaries in material, and crystals all tend to minimize energy and they often have interesting singularities. The study of such singularities leads to various subproblems, including questions of combinatorial geometry.¹