A decomposition theorem for balanced measures

Let $G=(V,E)$ be a connected graph. A probability measure $\mu$ on $V$ is called “balanced” if it has the following property: if $T_{\mu }(v)$ denotes the “earth mover’s” cost of transporting all the mass of $\mu$ from all over the graph to the vertex $v$, then $T_{\mu }$ attains its global maximum at each point in the support of $\mu$. 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.