A decomposition theorem for balanced measures

Let $G=(V,E)$ be a connected graph. A probability measure μ on V is called balanced if it has the following property: if $T_{\mu }\left(v\right)$ denotes the “earth mover's” cost of transporting all the mass of μ from all over the graph to the vertex v, then $T_{\mu }$ attains its global maximum at each point in the support of μ. We prove a decomposition result that characterizes balanced measures as convex combinations of suitable “extremal” balanced measures that we call basic. An upper bound on the number of basic balanced measures on G follows, and an example shows that this estimate is essentially sharp.