论文标题
从翻译不变估值的谐波分析到凸体的几何不平等现象
From harmonic analysis of translation-invariant valuations to geometric inequalities for convex bodies
论文作者
论文摘要
Alesker-Bernig-Schuster定理断言，特殊正交组的每个不可约表示，最多都以多种性能作为固定程度同质性的连续翻译不变估值的空间的子代表。此外，定理用最高权重描述，这些权重以多种表现出来。在本文中，我们介绍了该结果的改进，即在每个不可约亚代表性中明确构造最高权重矢量。然后，我们描述自然操作对估值的重要性（撤退，推动力，傅立叶变换，lefschetz运营商，艾莱斯克皇帝配对）对这些最高权重向量的作用。我们使用这些信息来证明霍奇 - 里曼的关系是欧几里得球作为参考机构的估值。由于最近已使用特殊案例 - 里奇 - 里曼关系来证明凸体的新几何不平等现象，因此我们的工作立即扩大了这些不平等的范围。
The Alesker-Bernig-Schuster theorem asserts that each irreducible representation of the special orthogonal group appears with multiplicity at most one as a subrepresentation of the space of continuous translation-invariant valuations with fixed degree of homogeneity. Moreover, the theorem describes in terms of highest weights which irreducible representations appear with multiplicity one. In this paper, we present a refinement of this result, namely the explicit construction of a highest weight vector in each irreducible subrepresentation. We then describe how important natural operations on valuations (pullback, pushforward, Fourier transform, Lefschetz operator, Alesker-Poincaré pairing) act on these highest weight vectors. We use this information to prove the Hodge-Riemann relations for valuations in the case of Euclidean balls as reference bodies. Since special cases of the Hodge-Riemann relations have recently been used to prove new geometric inequalities for convex bodies, our work immediately extends the scope of these inequalities.
