Log-Submodularity of Zonoid Volume
Statement
The conjecture that volume is log-submodular under Minkowski addition on zonoids, that is |A||A+B+C| <= |A+B||A+C|. Disproved by a four-dimensional zonotope generated by a 2-modular matrix together with two segments. Several related local mixed-volume, local Loomis-Whitney, projection-volume-ratio and volume-to-surface-area conjectures fall with it.
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No person has examined this. Nothing here has been checked at all. say whether it holds →
construction · #1
Ruben Skorupinski, using GPT-5.6 ProThat credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.
The paper has a section called How the counterexample was found. The author proved the unimodular case and identified 2-modular matrices as the place to look; then "Chat-GPT was then used to search for a counterexample within the space of the 2-modular matrices which led to the discovery of the counterexample within a specific class of 2-modular matrices". The acknowledgment is more conservative, crediting GPT-5.6 Pro with "literature searches and exploratory volume computations of 2-modular zonotopes", all independently verified by the author. Classified on the lower of the two readings.
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