A geometric inequality with applications to linear forms

Let CN be a cube of volume one centered at the origin in RN and let Pκ be a iΓ-dimensional subspace of RN. We prove that CN Π Pκ has iΓ-dimensionai volume greater than or equal to one. As an application of this inequality we obtain a precise version of Minkowski's linear forms theorem. We also state a conjecture which would allow our method to be generalized. I* Introduction* Let CN — [ — 1/2, 1/2]^ be the iV-dimensional cube of volume one centered at the origin in RN and suppose that Pκ is a if-dimensional linear subspace of RN. Dr. Anton Good has conjectured that the iί-dimensional volume of Pκ Π CN is always greater than or equal to one. In case K = N — 1 this has recently been proved by Hensley [6], who also obtained upper bounds for this volume. Our purpose in this paper is to prove the conjecture for arbitrary K and to give some applications to Minkowski's theorem on linear forms. In fact we prove a more general inequality for the product of spheres of various dimensions which contains the conjecture as a special case. We write x for the column vector I I in Rn and