Central diagonal sections of the $n$-cube
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866910154855809024 |
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| author | Bartha, Ferenc Fodor, Ferenc Merino, Bernardo González |
| author_facet | Bartha, Ferenc Fodor, Ferenc Merino, Bernardo González |
| contents | We prove that the volume of central hyperplane sections of a unit cube in $\mathbb{R}^n$ orthogonal to a diameter of the cube is a strictly monotonically increasing function of the dimension for $n\geq 3$. Our argument uses an integral formula that goes back to Pólya \cite{P} (see also \cite{H} and \cite{B86}) for the volume of central sections of the cube, and Laplace's method to estimate the asymptotic behaviour of the integral. First we show that monotonicity holds starting from some specific $n_0$. Then, using interval arithmetic (IA) and automatic differentiation (AD), we compute an explicit bound for $n_0$, and check the remaining cases between $3$ and $n_0$ by direct computation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2005_08292 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Central diagonal sections of the $n$-cube Bartha, Ferenc Fodor, Ferenc Merino, Bernardo González Metric Geometry Functional Analysis We prove that the volume of central hyperplane sections of a unit cube in $\mathbb{R}^n$ orthogonal to a diameter of the cube is a strictly monotonically increasing function of the dimension for $n\geq 3$. Our argument uses an integral formula that goes back to Pólya \cite{P} (see also \cite{H} and \cite{B86}) for the volume of central sections of the cube, and Laplace's method to estimate the asymptotic behaviour of the integral. First we show that monotonicity holds starting from some specific $n_0$. Then, using interval arithmetic (IA) and automatic differentiation (AD), we compute an explicit bound for $n_0$, and check the remaining cases between $3$ and $n_0$ by direct computation. |
| title | Central diagonal sections of the $n$-cube |
| topic | Metric Geometry Functional Analysis |
| url | https://arxiv.org/abs/2005.08292 |