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 ${\mathbb{R}}^n$, and let $F$ be an affine $(n-d)$-dimensional subspace of ${\mathbb{R}}^n$ having distance to the origin less than or equal to $\frac{1}{2}$, where $0<d<n$. We show that the $(n-d)$-dimensional volume of the section $Q_n \cap F$isboundedbelowbyavalue$c(d)$dependingonlyonthecodimension$d$butnotontheambientdimension$n$oraparticularsubspace$F$. In the case of hyperplanes, $d=1$, we show that $c(1)=\frac{1}{17}$ is a possible choice. We also consider a complex analogue of this problem for a hyperplane section of the polydisc.