A geometric atudy of the split decomposition

Abstract

This paper sheds new light on split decomposition theory and T-theory from the viewpoint of convex analysis and polyhedral geometry. By regarding finite metrics as discrete concave functions, Bandelt-Dress' split decomposition can be derived as a special case of more general decomposition of polyhedral/discrete concave functions introduced in this paper. It is shown that the combinatorics of splits discussed in connection with the split decomposition corresponds to the geometric properties of a hyperplane arrangement and a point configuration. Using our approach, the split decomposition of metrics can be naturally extended to distance functions, which may violate the triangle inequality, using partial split distances. © 2006 Springer Science+Business Media, Inc.