Résumé : It is well known that a line can intersect at most $2n-1$ cells of the $n \times n$ chessboard. What happens in higher dimensions: how many cells of the $d$-dimensional $[0,n]^d$ box can a hyperplane intersect? We determine this number asymptotically. We also prove the integer analogue of the following fact. If $K,L$ are convex bodies in $R^d$ and $K \subset L$, then the surface area $K$ is smaller than that of $L$. Joint work with Peter Frankl.
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