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arXiv · 2005.08292

Central diagonal sections of the $n$-cube

Ferenc Bartha·Ferenc Fodor·Bernardo González Merino
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Abstract

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 P\'olya \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.

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Ferenc Bartha, Ferenc Fodor, Bernardo González Merino. 2020-05-17. Central diagonal sections of the $n$-cube. https://doi.org/10.1093/imrn%2Frnaa254

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