体积与投影不等式I：带状体与科塔德猜想
Volume and Projection Inequalities I: Zonoids and Courtade's Conjecture
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中文总结 AI 辅助
该论文研究带状体的体积与投影不等式，证明对数次模性猜想等价于zonotope体积多项式的瑞利性质，在次数/余次数≤3时成立，次数/余次数≥4时构造反例，还否定了科塔德猜想在三维及以上带状体中的成立性。
中文摘要 AI 辅助
我们通过编码带状体体积的多重仿射行列式多项式，研究带状体的体积与投影不等式。我们证明，关于带状体体积的对数次模性猜想，等价于zonotope体积多项式的瑞利性质，且我们在次数或余次数至多为3的情形下证明了该性质。该结果是尖锐的：当次数和余次数至少为4时，我们利用秩至少为4的非瑞利拟阵的存在性构造了反例。此外，我们在四维及更高维中提供了该猜想等价的投影不等式形式的幺模或图反例，还证明了更强的投影不等式在三维中已不成立。我们还利用一对正交的旋转双体否定了科塔德猜想，尽管该猜想最初针对一般凸体提出，但我们证明其在每维至少为3的带状体上均不成立。
英文摘要
We study volume and projection inequalities for zonoids through the multiaffine determinant polynomials that encode their volumes. We show that a log-submodularity conjecture for the volume of zonoids is equivalent to the Rayleigh property of zonotope volume polynomials, which we prove for the case when the degree or codegree is at most 3. This result is sharp in that when the degree and codegree are at least 4, we construct counterexamples using the existence of non-Rayleigh matroids in ranks at least four. Additionally, we provide unimodular or graphical counterexamples in dimensions four and higher to an equivalent projection inequality formulation of the conjecture. We also show that a stronger projection inequality fails already in dimension three. We next disprove Courtade's conjecture using a pair of orthogonal double bodies of revolution. Although Courtade's conjecture was originally formulated for general convex bodies, we show that it fails even for zonoids in every dimension at least three.
