On the volume of non-central sections of a cube

Let Q(n) be the cube of side length one centered at the origin in R-n, and let F be an affine (n - d)-dimensional subspace of R-n having distance to the origin less than or equal to 1/2 where 0 < d < n. We show that the (n - d)-dimensional volume of the section Q(n)boolean AND 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) = 1/17 is a possible choice. We also consider a complex analogue of this problem for a hyperplane section of the polydisc. (C) 2019 Elsevier Inc. All rights reserved.