It is well known that a line can intersect at most 2n−1 cells of the n×n chessboard. What happens in higher dimensions: how many cells of the d-dimensional [0,n]^^d box can a hyperplane intersect? We answer this question asymptotically. We also prove the integer analogue of the following fact. If K,L are convex bodies in R^d and K ⊂ L, then the surface area K is smaller than that of L. This is joint work with Péter Frankl.