Uniform cover inequalities for the volume of coordinate sections and projections of convex bodies

摘要

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