Cells in the box and a hyperplane

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 $n\times \ldots \times n$ box can a hyperplane intersect? We also prove the lattice analogue of the following well-known fact. If $K,L$ are convex bodies in $R^d$ and $K\subset L$, then the surface area of $K$ is smaller than that of $L$.