A Note on Circular Decomposable Metrics

A metric d on a finite set X is called a Kalmanson metric if there exists a circular ordering of points of X, such that d(y, u) + d(z, v) d(y, z) + d(u, v) for all crossing pairs yu and zv of . We prove that any Kalmanson metric d is an l₁-metric, i.e. d can be written as a nonnegative linear combination of split metrics. The splits in the decomposition of d can be selected to form a circular system of splits in the sense of Bandelt and Dress.