Bisections of mass assignments by parallel hyperplanes
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916837844844544 |
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| author | Sadovek, Nikola Soberón, Pablo |
| author_facet | Sadovek, Nikola Soberón, Pablo |
| contents | In this paper, we prove a result on the bisection of mass assignments by parallel hyperplanes on Euclidean vector bundles. Our methods consist of the development of a novel lifting method to define the configuration space--test map scheme, which transforms the problem to a Borsuk--Ulam-type question on equivariant fiber bundles, along with a new computation of the parametrized Fadell--Husseini index. As the primary application, we show that any $d+k+m-1$ mass assignments to linear $d$-spaces in $\mathbb{R}^{d+m}$ can be bisected by $k$ parallel hyperplanes in at least one $d$-space, provided that the Stirling number of the second kind $S(d+k+m-1, k)$ is odd. This generalizes all known cases of a conjecture by Soberón and Takahashi, which asserts that any $d+k-1$ measures in $\mathbb{R}^d$ can be bisected by $k$ parallel hyperplanes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04058 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bisections of mass assignments by parallel hyperplanes Sadovek, Nikola Soberón, Pablo Algebraic Topology 55N91, 52C35, 52A37, 55R91 In this paper, we prove a result on the bisection of mass assignments by parallel hyperplanes on Euclidean vector bundles. Our methods consist of the development of a novel lifting method to define the configuration space--test map scheme, which transforms the problem to a Borsuk--Ulam-type question on equivariant fiber bundles, along with a new computation of the parametrized Fadell--Husseini index. As the primary application, we show that any $d+k+m-1$ mass assignments to linear $d$-spaces in $\mathbb{R}^{d+m}$ can be bisected by $k$ parallel hyperplanes in at least one $d$-space, provided that the Stirling number of the second kind $S(d+k+m-1, k)$ is odd. This generalizes all known cases of a conjecture by Soberón and Takahashi, which asserts that any $d+k-1$ measures in $\mathbb{R}^d$ can be bisected by $k$ parallel hyperplanes. |
| title | Bisections of mass assignments by parallel hyperplanes |
| topic | Algebraic Topology 55N91, 52C35, 52A37, 55R91 |
| url | https://arxiv.org/abs/2412.04058 |