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arXiv · 2312.08649

A decomposition theorem for balanced measures

Gregory Baimetov·Ryan Bushling·Ansel Goh·Raymond Guo·Owen Jacobs·Sean Lee
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Abstract

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.

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Gregory Baimetov, Ryan Bushling, Ansel Goh, Raymond Guo, Owen Jacobs, Sean Lee. 2023-12-14. A decomposition theorem for balanced measures. https://doi.org/10.1016/j.disc.2024.114389

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