Shallow sections of the hypercube

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.