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| Main Authors: | , , , , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.08649 |
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| _version_ | 1866915095239458816 |
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| author | Baimetov, Gregory Bushling, Ryan Goh, Ansel Guo, Raymond Jacobs, Owen Lee, Sean |
| author_facet | Baimetov, Gregory Bushling, Ryan Goh, Ansel Guo, Raymond Jacobs, Owen Lee, Sean |
| contents | Let $G = (V,E)$ be a connected graph. A probability measure $μ$ on $V$ is called "balanced" if it has the following property: if $T_μ(v)$ denotes the "earth mover's" cost of transporting all the mass of $μ$ from all over the graph to the vertex $v$, then $T_μ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_08649 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A decomposition theorem for balanced measures Baimetov, Gregory Bushling, Ryan Goh, Ansel Guo, Raymond Jacobs, Owen Lee, Sean Combinatorics 05C12, 05C69 Let $G = (V,E)$ be a connected graph. A probability measure $μ$ on $V$ is called "balanced" if it has the following property: if $T_μ(v)$ denotes the "earth mover's" cost of transporting all the mass of $μ$ from all over the graph to the vertex $v$, then $T_μ$ 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. |
| title | A decomposition theorem for balanced measures |
| topic | Combinatorics 05C12, 05C69 |
| url | https://arxiv.org/abs/2312.08649 |