Deep sections of the hypercube

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.