Deep sections of the hypercube

TL;DR

This paper investigates how the volume of hyperplane sections of the hypercube changes with dimension and position, revealing conditions for monotonicity and extremality, and providing convergence estimates for Eulerian numbers.

Contribution

It extends the understanding of the monotonicity and extremality of hypercube sections for various positions and dimensions, and offers new bounds and convergence rates.

Findings

Volume increases with dimension for certain hyperplanes when t=0.

Identifies ranges of t where volume decreases with dimension.

Shows large d and t conditions for local maximality of the volume.

Abstract

Consider a non-negative number $t$ and a hyperplane $H$ of $\mathbb{R}^d$ whose distance to the center of the hypercube $[0,1]^d$ is $t$. If $t$ is equal to $0$ and $H$ is orthogonal to a diagonal of $[0,1]^d$, it is known that the $(d-1)$-dimensional volume of $H\cap[0,1]^d$ is a strictly increasing function of $d$ when $d$ is at least $3$. The study of the monotonicity of this volume is extended for $t$ up to above $1/2$ and, when $d$ is large enough, for every non-negative $t$. In particular, a range for $t$ is identified such that this volume is a strictly decreasing function of $d$ over the positive integers. The local extremality of the $(d-1)$-dimensional volume of $H\cap[0,1]^d$ when $H$ is orthogonal to a diagonal of either $[0,1]^d$ or a lower dimensional face is also determined for the same values of $t$. It is shown for instance that when $t$ is above an explicit constant and $d$ is large enough, this volume is always strictly locally maximal when $H$ is orthogonal to a diagonal of $[0,1]^d$. A precise estimate for the convergence rate of the Eulerian numbers to their limit Gaussian behavior is provided along the way.