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

TL;DR

This paper extends classical volume inequalities for convex bodies, providing new bounds for coordinate projections and sections, generalizing Loomis-Whitney and Meyer's dual inequalities.

Contribution

It introduces new uniform cover inequalities for convex bodies, extending existing bounds to coordinate sections and projections with dual formulations.

Findings

Established new lower bounds for volume via coordinate projections.

Derived dual inequalities for coordinate sections of convex bodies.

Extended classical inequalities to broader convex geometric settings.

Abstract

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.