Bisecting masses with families of parallel hyperplanes
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911857806147584 |
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| author | Hubard, Alfredo Soberón, Pablo |
| author_facet | Hubard, Alfredo Soberón, Pablo |
| contents | We prove a common generalization to several mass partition results using hyperplane arrangements to split $\mathbb{R}^d$ into two sets. Our main result implies the ham-sandwich theorem, the necklace splitting theorem for two thieves, a theorem about chessboard splittings with hyperplanes with fixed directions, and all known cases of Langerman's conjecture about equipartitions with $n$ hyperplanes.
Our main result also confirms an infinite number of previously unknown cases of the following conjecture of Takahashi and Soberón:
For any $d+k-1$ measures in $\mathbb{R}^d$, there exist an arrangement of $k$ parallel hyperplanes that bisects each of the measures.
The general result follows from the case of measures that are supported on a finite set with an odd number of points. The proof for this case is inspired by ideas of differential and algebraic topology, but it is a completely elementary parity argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_14320 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bisecting masses with families of parallel hyperplanes Hubard, Alfredo Soberón, Pablo Combinatorics 52C35, 52A37, 28A75 We prove a common generalization to several mass partition results using hyperplane arrangements to split $\mathbb{R}^d$ into two sets. Our main result implies the ham-sandwich theorem, the necklace splitting theorem for two thieves, a theorem about chessboard splittings with hyperplanes with fixed directions, and all known cases of Langerman's conjecture about equipartitions with $n$ hyperplanes. Our main result also confirms an infinite number of previously unknown cases of the following conjecture of Takahashi and Soberón: For any $d+k-1$ measures in $\mathbb{R}^d$, there exist an arrangement of $k$ parallel hyperplanes that bisects each of the measures. The general result follows from the case of measures that are supported on a finite set with an odd number of points. The proof for this case is inspired by ideas of differential and algebraic topology, but it is a completely elementary parity argument. |
| title | Bisecting masses with families of parallel hyperplanes |
| topic | Combinatorics 52C35, 52A37, 28A75 |
| url | https://arxiv.org/abs/2404.14320 |