zonoid体积不是对数次模性的
Zonoid volumes are not log-submodular
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中文总结 AI 辅助
该研究构造反例证明zonoid类上体积的对数次模性猜想不成立，相关多个几何猜想也随之失效，同时给出单模zonotope情形下该猜想成立的证明及等号刻画。
中文摘要 AI 辅助
我们通过构造一个四维zonotope A以及两个线段B和C，证明了关于zonoid类上Minkowski加法下体积是对数次模性的猜想不成立，满足|A||A+B+C|>|A+B||A+C|，其中|·|表示体积。因此，多个相关的局部混合体积、局部Loomis-Whitney、投影体积比及体积表面积比猜想对zonoid也不成立。该反例中的zonotope由一个2-模矩阵生成，我们还证明了当A+B+C是单模zonotope时该猜想成立，并给出了此情形下的等号情况刻画。
英文摘要
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.
