geometry-topology / Convex geometry
The conjecture that volume is log-submodular under Minkowski addition on zonoids, that is $|A||A+B+C| \leq |A+B||A+C|$. 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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The paper also proves the conjecture in the unimodular case and characterizes equality there, so the boundary between true and false is drawn rather than just crossed.
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The conjecture that volume is log-submodular under Minkowski addition on zonoids, that is $|A||A+B+C| \leq |A+B||A+C|$. 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.
The paper also proves the conjecture in the unimodular case and characterizes equality there, so the boundary between true and false is drawn rather than just crossed.
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