VOLUMES OF PROJECTIONS OF UNIT CUBES PETER McMULLEN ABSTRACT Let Z and 2 be the images of a regular unit n-cube in E" under orthogonal projection on to orthogonal complementary subspaces of dimensions d and n — d respectively. It is shown that the d-volume of Z and the (« — d)-volume of Z are equal. This generalizes to a connexion between the volumes of strongly associated zonotopes. 1. Introduction In this paper, we shall show that a regular unit n-cube in E" has a certain curious property. Denoting r-dimensional volume (of r-dimensional compact convex sets) by V r , we shall prove THEOREM 1. Let Z and Z be the images of a regular unit n-cube in E" under orthogonal projection on to orthogonal complementary subspaces of E" of dimensions d and n — d respectively. Then V d {Z) = V n _ d (Z). Now Z and Z are zonotopes (Minkowski sums of line segments); indeed, they are associated zonotopes in the sense of [2] or [5]. Our proof of Theorem 1 will involve properties of associated zonotopes, and we shall see that the theorem can be generalized to arbitrary pairs of zonotopes, which are associated in a strong sense. We wish to thank the referee for drawing attention to Jacobi's theorem, used to prove Lemma 5 below. 2. Associated zonotopes In this section, we recall those properties of zonotopes which we shall use in proving Theorem 1. Let Z be a ^-dimensional zonotope (or d-zonotope for short) in E d . Thus Z is a vector or Minkowski sum of line segments, say where Sj = [afij] = cornea,-, b } ) (j = l,...,n). As usual (see [2], [3] or [5] for general references to what follows), we allow constituent line segments Sj to be parallel, or even to degenerate to points. Zonotopes are clearly centrally symmetric, and so it is customary to take their centres to be the origin o of coordinates; thus each line segment Sj can be taken to have the form S, = \_( — Xj)xj]. However, we shall find it convenient here to adopt a different convention, and take Sj to be of the form Sj = \_0Xj] (j = l,...,«). Received 7 September, 1983. Bull. London Math. Soc, 16 (1984), 278-280