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 $Q_n \cap F$ 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 $c(1) = \frac{1}{17}$ is a possible choice. We also consider a complex analogue of this problem for a hyperplane section of the polydisc.