Deep sections of the hypercube
Lionel Pournin
TL;DR
This work analyzes the volumes of hyperplane sections $H\cap[0,1]^d$ of the unit hypercube when $H$ is at distance $t$ from the center and, in particular, orthogonal to a diagonal. It derives a sharp second-order expansion for the diagonal-orthogonal volumes $I_d(t)$, establishing precise Gaussian convergence rates and linking them to Eulerian-number asymptotics; it then studies how these volumes change with dimension $d$, identifying t-intervals where $I_d(t)$ is increasing or decreasing in $d$. The results yield explicit thresholds and asymptotics for local extremality of diagonal sections, showing strict local maximality for many regimes and non-extremality for sections corresponding to high-dimensional faces, with rigorous bounds and symbolic-computation verifications. By tying hypercube-section geometry to Eulerian numbers and Gaussian limits, the paper provides a comprehensive picture of how high-dimensional diagonally aligned hypercube sections behave, including both global monotonicity and local extremality properties.
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.