This paper sheds a new light on the 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 to 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
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