On the volume of non-central sections of a cube

Abstract

Let Qn be the cube of side length one centered at the origin in Rn, and let F be an affine (n−d)-dimensional subspace of Rn having distance to the origin less than or equal to 12, where 0<d<n. We show that the (n−d)-dimensional volume of the section Qn∩F is bounded below by a value c(d) depending only on the codimension d but not on the ambient dimension n or a particular subspace F. In the case of hyperplanes, d=1, we show that c(1)=117 is a possible choice. We also consider a complex analogue of this problem for a hyperplane section of the polydisc.