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.

Disclosure

“|A + B| |A| ≥ . |∂(A + B)|d−1 |∂A|d−1 4 Acknowledgments GPT-5.6 Pro was used during the development of this paper to aid with literature searches and with exploratory volume computations of 2-modular zonotopes. All computations were independently verified by the author. References [1] Stephan Ar”

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Classification

Computational experiments or data processing

Multiplier

3

Verified

2026-08-23T05:00:22.254507Z

Structural counts