Central diagonal sections of the $n$-cube

TL;DR

This paper proves that the volume of central hyperplane sections of a unit cube, orthogonal to a diameter, increases monotonically with dimension for all sufficiently large n.

Contribution

It establishes the monotonicity of the volume function for all dimensions n ≥ 3 using integral formulas, asymptotic analysis, and computational methods.

Findings

Monotonicity holds for all n ≥ 3 starting from some n0.

Explicit bounds for n0 are computed using interval arithmetic and automatic differentiation.

The remaining cases are verified through direct computation.

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.