A note on M-convexity in polyhedral split decomposition of distances

Abstract

Buneman’s decomposition of a metric can be obtained in a geometric way as the polyhedral split decomposition of a polyhedral convex function associated with a metric. In discrete convex analysis, such a polyhedral convex function is considered as an important example of M-convex functions. Motivated by this, we shed light on the decomposition from the viewpoint of discrete convex analysis. We firstly show that the decomposition is a sum of M-convex functions, where a sum of M-convex functions is not necessarily M-convex in general. We explain why the M-convexity is preserved in the decomposition. We next show that a quadratic M-convex function is split-decomposable at every point. This indicates simplicity of the geometric structure of a quadratic M-convex function. We finally give a characterization of M-convex polyhedra in terms of their facets. This can be used to describe the M-convexity of a polyhedron associated with a function on a cross-free family.