  A theorem of W. Derrick ensures that the volume of any Riemannian cube
$([0,1]^n,g)$ is bounded below by the product of the distances between opposite
codimension-1 faces. In this paper, we establish a discrete analog of Derrick's
inequality for weighted open covers of the cube $[0,1]^n$, which is motivated
by a question about lower volume bounds in metric spaces. Our main theorem
generalizes a previous result of the author, which gave a combinatorial version
of Derrick's inequality and was used in the analysis of boundaries of
hyperbolic groups. As an application, we answer a question of Y. Burago and V.
Zalgaller about length-volume inequalities for pseudometrics on the unit cube.
