Non-diagonal critical central sections of the cube

TL;DR

This paper investigates the volume of central hyperplane sections of high-dimensional cubes, providing new proofs for maximal sections and establishing the existence of non-diagonal critical sections in all dimensions at least four.

Contribution

It offers a simpler proof for the maximality of sections perpendicular to the main diagonal and proves the existence of non-diagonal critical sections in all dimensions ≥4.

Findings

Main diagonal sections are strictly locally maximal for n ≥ 4.

Non-diagonal critical sections exist in all dimensions ≥4.

Derived bounds for Eulerian numbers of the first kind.

Abstract

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-P\'olya 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.