A corner cut in dimension d is a finite subset of Nd0 that can be separated from its complement in Nd0 by an affine hyperplane disjoint from N d 0 . Corner cuts were first investigated by Onn and Sturmfels , the original motivation stemming from computational commutative algebra. Let us write N d 0 k cut for the set of corner cuts of cardinality k; in the computational geometer’s terminology, these are the k-sets of Nd0. Among other things, Onn and Sturmfels give an upper bound of O(k2d d−1 d+1 ) for the size of N d 0 k cut when the dimension is fixed. In two dimensions, it is known (see ) that # N d 0 k cut = Θ(k log k). We will see that in general, for any fixed dimension d, the order of magnitude of # N d 0 k cut is between k log k and (k log k). (It has been communicated to me that the same bounds have been found independently by Gaël Rémond.) In fact, the elements of N d 0 k cut correspond to the vertices of a certain polytope, and what our proof will show is that the above upper bound holds for the total number of flags of that polytope.