Cells in the box and a hyperplane

Abstract

It is well known that a line can intersect at most $2n-1$ cells of the $n\times n$ chessboard. Here we consider the high-dimensional version: how many cells of the $d$-dimensional $n\times \cdots \times n$ box can a hyperplane intersect? We also prove the lattice analogue of the following well-known fact: if $K,L$ are convex bodies in ${\mathbb{R}}^d$ and $K\subset L$, then the surface area of $K$ is smaller than that of $L$.