For any set H of n hyperplanes in R^k, and any parameter r, 1 ≤ r≤ n, there always exists a (1/r)-cutting of size O(r^k). In two dimensions, a (1/r)-cutting of size s is a partition of the plane into s disjoint triangles, some of which are unbounded, such that no triangle in the partition intersects more than n/r lines in H. In R^k, triangles are replaced by simplices. Such a cutting can be computed in O(nr^k-1) time.