Central diagonal sections of the $n$-cube

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 $n\ge 3$. 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.

16 pages

Cited by in corpus (3)

• Stability of polydisc slicing
• Non-diagonal critical central sections of the cube
• Estimates on the decay of the Laplace-Polya integral