  The classical Loomis-Whitney inequality and the uniform cover inequality of
Bollob\'{a}s and Thomason provide lower bounds for the volume of a compact set
in terms of its lower dimensional coordinate projections. We provide further
extensions of these inequalities in the setting of convex bodies. We also
establish the corresponding dual inequalities for coordinate sections; these
uniform cover inequalities for sections may be viewed as extensions of Meyer's
dual Loomis-Whitney inequality.
