On the volume of sums of anti-blocking bodies

TL;DR

This paper investigates volume inequalities for sums of anti-blocking bodies, establishing new bounds and inequalities, including analogues of classical results, with sharp constants, and extending to unconditional product measures.

Contribution

It introduces novel volume inequalities for anti-blocking bodies, including analogues of Pl"unnecke-Ruzsa and Milman inequalities, with sharp constants, and extends results to unconditional product measures.

Findings

Proved Pl"unnecke-Ruzsa type inequalities for anti-blocking bodies.

Established Milman inequalities on volume ratios.

Extended inequalities to unconditional product measures.

Abstract

We study inequalities on the volume of Minkowski sum in the class of anti-blocking bodies. We prove analogues of Pl\"unnecke-Ruzsa type inequality and V. Milman inequality on the concavity of the ratio of volumes of bodies and their projections. We also study Firey $L_p-$sums of anti-blocking bodies and prove Pl\"unnecke-Ruzsa type inequality; V. Milman inequality and Roger-Shephard inequality. The sharp constants are provided in all of those inequalities, for the class of anti-blocking bodies. Finally, we extend our results to the case of unconditional product measures with decreasing density.