The Kantorovich metric for probability measures on the circle

In this paper we show that there exists an analytic expression for the Kantorovich distance between probebility measures on the circle. Previously such an expression was only known for measures supported on the real line. In the case that the measures are discrete, this formula enables us to show that the Kantorovich distance can be computed in linear time. This is important for applications, in particular in pattern recognition where this distance is used for texture analysis. As another application we see that the analytic expression found allows us to solve a Minimal Matching Problem in linear time, for which so far only n log n algorithms were known.