Shallow sections of the hypercube

TL;DR

This paper investigates the maximal $(d-1)$-dimensional volume of hyperplane sections of a hypercube under specific geometric constraints, extending known results to higher dimensions with relaxed conditions.

Contribution

It generalizes previous findings by showing the maximal hyperplane sections occur when the hyperplane is orthogonal to a hypercube diagonal, even under weaker containment conditions for the ball.

Findings

Maximal hyperplane sections occur when hyperplanes are orthogonal to a hypercube diagonal for $d ext{ } extgreater{} ext{ }4$.

The result extends known geometric properties to higher dimensions with less restrictive conditions.

The geometric configuration involves a ball containing face centers but excluding vertices and edge midpoints.

Abstract

Consider a $d$-dimensional closed ball $B$ whose center coincides with that of the hypercube $[0,1]^d$. Pick the radius of $B$ in such a way that the vertices of the hypercube are outside of $B$ and the midpoints of its edges in the interior of $B$. It is known that, when $d\geq3$, the $(d-1)$-dimensional volume of $H\cap[0,1]^d$, where $H$ is a hyperplane of $\mathbb{R}^d$ tangent to $B$, is largest possible if and only if $H$ is orthogonal to a diagonal of the hypercube. It is shown here that the same holds when $d\geq5$ but the interior of $B$ is only required to contain the centers of the square faces of the hypercube.