Geometric Inequalities for Anti-Blocking Bodies

TL;DR

This paper establishes new geometric inequalities for anti-blocking convex bodies, including bounds on volumes and mixed volumes, and applies these results to combinatorics of posets, advancing understanding in convex geometry and combinatorics.

Contribution

It proves several conjectures and inequalities for anti-blocking bodies, extending classical results and introducing novel decompositions applicable to general polyhedral cones.

Findings

Proved Godberson's conjecture for anti-blocking bodies.

Derived near-optimal bounds on Mahler volumes.

Established Sidorenko-type inequalities for poset linear extensions.

Abstract

We study the class of (locally) anti-blocking bodies as well as some associated classes of convex bodies. For these bodies, we prove geometric inequalities regarding volumes and mixed volumes, including Godberson's conjecture, near-optimal bounds on Mahler volumes, Saint-Raymond-type inequalities on mixed volumes, and reverse Kleitman inequalities for mixed volumes. We apply our results to the combinatorics of posets and prove Sidorenko-type inequalities for linear extensions of pairs of 2-dimensional posets. The results rely on some elegant decompositions of differences of anti-blocking bodies, which turn out to hold for anti-blocking bodies with respect to general polyhedral cones.