A note on circular decomposable metrics (Q1381337)

The authors consider a finite metric space \(X\) with metric \(d\). If there is a circular ordering of points of \(X\) such that \(d(y,u)+ d(z,v)\geq d(y,z)+ d(u,v)\), for all crossing pairs \(yu\) and \(zv\), then \(d\) is called a Kalmanson metric. The paper gives a new and shorter proof that \(d\) is a Kalmanson metric if and only if \(d\) is circularly decomposable. Originally, this theorem was proved by \textit{G. Christopher, M. Farach} and \textit{M. A. Trick} [The structure of circular decomposable metrics, Lect. Notes Comput. Sci. 1136, 486-500 (1996)]. The present paper gives also the following corollary. If \(d\) is a convex metric on a topological two-disc bounded by a Jordan curve \(\Gamma\), then the metric space \((\Gamma,d)\) is \(L_1\)-embeddable. Recall that \(d\) is called a convex metric if for every distinct points \(x,y\in X\) there exists a point \(z\in X\) different from \(x\) and \(y\) such that \(d(x,z)+ d(z,y)= d(x,y)\).