Zonoid volumes are not log-submodular

Abstract

We disprove the conjecture that volume is log-submodular under Minkowski addition on the class of zonoids by exhibiting a four-dimensional zonotope $A$ and two segments $B$ and $C$ such that \begin{align*} |A||A+B+C|>|A+B||A+C|, \end{align*} where $|\cdot |$ denotes the volume. Consequently, several related local mixed-volume, local Loomis-Whitney, projection-volume ratio, and volume-to-surface-area conjectures also fail for zonoids. The zonotope in our counterexample is generated by a 2-modular matrix. We also provide a proof of the correctness of the conjecture in the case where $A+B+C$ is a unimodular zonotope as well as a characterization of the equality cases in this setting.