On polynomial inequalities for cone-volumes of polytopes
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916799319113728 |
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| author | Baumbach, Tom Henk, Martin |
| author_facet | Baumbach, Tom Henk, Martin |
| contents | Motivated by the discrete logarithmic Minkowski problem we study for a given matrix $U\in\mathbb{R}^{n\times m}$ its cone-volume set $C_{\tt cv}(U)$ consisting of all the cone-volume vectors of polytopes $P(U,b)=\{ x\in\mathbb{R}^n : U^\intercal x\leq b\}$, $b\in\mathbb{R}^n_{\geq 0}$. We will show that $C_{\tt cv}(U)$ is a path-connected semialgebraic set which extends former results in the planar case or for particular polytopes. Moreover, we define a subspace concentration polytope $P_{\tt scc}(U)$ which represents geometrically the subspace concentration conditions for a finite discrete Borel measure on the sphere. This is up to a scaling the basis matroid polytope of $U$, and these two sets, $P_{\tt scc}(U)$ and $C_{\tt cv}(U)$, also offer a new geometric point of view to the discrete logarithmic Minkowski problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_15370 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On polynomial inequalities for cone-volumes of polytopes Baumbach, Tom Henk, Martin Metric Geometry Motivated by the discrete logarithmic Minkowski problem we study for a given matrix $U\in\mathbb{R}^{n\times m}$ its cone-volume set $C_{\tt cv}(U)$ consisting of all the cone-volume vectors of polytopes $P(U,b)=\{ x\in\mathbb{R}^n : U^\intercal x\leq b\}$, $b\in\mathbb{R}^n_{\geq 0}$. We will show that $C_{\tt cv}(U)$ is a path-connected semialgebraic set which extends former results in the planar case or for particular polytopes. Moreover, we define a subspace concentration polytope $P_{\tt scc}(U)$ which represents geometrically the subspace concentration conditions for a finite discrete Borel measure on the sphere. This is up to a scaling the basis matroid polytope of $U$, and these two sets, $P_{\tt scc}(U)$ and $C_{\tt cv}(U)$, also offer a new geometric point of view to the discrete logarithmic Minkowski problem. |
| title | On polynomial inequalities for cone-volumes of polytopes |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2506.15370 |