On the volume of the Minkowski sum of zonoids

TL;DR

This paper investigates inequalities related to zonoids in convex geometry, establishing equivalences, confirming conjectures in specific dimensions, and exploring extensions in the $L_p$-Brunn-Minkowski theory.

Contribution

It demonstrates the equivalence of several key inequalities for zonoids, confirms conjectures in three dimensions, and extends results to the $L_p$ setting, especially for p=2.

Findings

Confirmed conjectures in ${ m R}^3$

Established an improved inequality in ${ m R}^2$

Extended results to the $L_p$-Brunn-Minkowski theory for p=2

Abstract

We explore some inequalities in convex geometry restricted to the class of zonoids. We show the equivalence, in the class of zonoids, between a local Alexandrov-Fenchel inequality, a local Loomis-Whitney inequality, the log-submodularity of volume, and the Dembo-Cover-Thomas conjecture on the monotonicity of the ratio of volume to the surface area. In addition to these equivalences, we confirm these conjectures in ${\mathbb R}^3$ and we establish an improved inequality in ${\mathbb R^2}$. Along the way, we give a negative answer to a question of Adam Marcus regarding the roots of the Steiner polynomial of zonoids. We also investigate analogous questions in the $L_p$-Brunn-Minkowski theory, and in particular, we confirm all of the above conjectures in the case $p=2$, in any dimension.