Non-diagonal critical central sections of the cube
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866917702123126784 |
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| author | Ambrus, Gergely Gárgyán, Barnabás |
| author_facet | Ambrus, Gergely Gárgyán, Barnabás |
| contents | We study the $(n-1)$-dimensional volume of central hyperplane sections of the $n$-dimensional cube $Q_n$. Our main goal is two-fold: first, we provide an alternative, simpler argument for proving that the volume of the section perpendicular to the main diagonal of the cube is strictly locally maximal for every $n \geq 4$, which was shown before by L. Pournin. Then, we prove that non-diagonal critical central sections of $Q_n$ exist in all dimensions at least $4$. The crux of both proofs is an estimate on the rate of decay of the Laplace-Pólya integral $J_n(r) = \int_{-\infty}^\infty \mathrm{sinc}^n t \cdot \cos (rt) \mathrm{d} t$ that is achieved by combinatorial means. This also yields improved bounds for Eulerian numbers of the first kind. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_03792 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Non-diagonal critical central sections of the cube Ambrus, Gergely Gárgyán, Barnabás Metric Geometry Combinatorics We study the $(n-1)$-dimensional volume of central hyperplane sections of the $n$-dimensional cube $Q_n$. Our main goal is two-fold: first, we provide an alternative, simpler argument for proving that the volume of the section perpendicular to the main diagonal of the cube is strictly locally maximal for every $n \geq 4$, which was shown before by L. Pournin. Then, we prove that non-diagonal critical central sections of $Q_n$ exist in all dimensions at least $4$. The crux of both proofs is an estimate on the rate of decay of the Laplace-Pólya integral $J_n(r) = \int_{-\infty}^\infty \mathrm{sinc}^n t \cdot \cos (rt) \mathrm{d} t$ that is achieved by combinatorial means. This also yields improved bounds for Eulerian numbers of the first kind. |
| title | Non-diagonal critical central sections of the cube |
| topic | Metric Geometry Combinatorics |
| url | https://arxiv.org/abs/2307.03792 |