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. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.