Hard Lefschetz theorem for valuations and related questions of integral geometry

Hard Lefschetz theorem for valuations, complex integral geometry, and unitarily invariant valuations

We obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets (a version of the hard Lefschetz theorem). Using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations. It implies new integral geometric formulas for real submanifolds in Hermitian spaces generalizing the classical kinematic formulas in Euclidean spaces due to Poincaré, Chern, Santaló, and others.

凸 valuations 的硬 Lefschetz 定理与 Hodge-Riemann 关系

Hard Lefschetz theorem and Hodge-Riemann relations for convex valuations

本文证明了凸体上光滑平移不变 valuations 的硬 Lefschetz 定理和 Hodge-Riemann 关系，并由此得到 McMullen 二次不等式对光滑严格正曲率边界凸体的成立。

凸赋值上的 Lefschetz 算子

Lefschetz operators on convex valuations

本文研究 Alesker 的 Lefschetz 算子作用于凸体上平移不变赋值的行为，对标量值赋值用 Radon 型变换刻画其作用，对旋转等变 Minkowski 赋值证明生成函数经卷积变换且卷积核满足 Legendre 型微分方程并严格正、除一点外光滑。

On Hodge-Riemann relations for translation-invariant valuations

The Alesker product turns the space of smooth translation-invariant valuations on convex bodies into a commutative associative unital algebra, satisfying Poincaré duality and the hard Lefschetz theorem. In this article, a version of the Hodge-Riemann relations for the Alesker algebra is conjectured, and the conjecture is proved in two particular situations: for even valuations, and for 1-homogeneous valuations. The latter result is then used to deduce a special case of the Aleksandrov-Fenchel inequality. Finally, mixed versions of the hard Lefschetz theorem and of the Hodge-Riemann relations a…

Algebraic structures on valuations, their properties and applications

We describe various structures of algebraic nature on the space of continuous valuations on convex sets, their properties (like versions of Poincaré duality and hard Lefschetz theorem), and their relations and applications to integral geometry.

Even Minkowski Valuations

A new integral representation of smooth translation invariant and rotation equivariant even Minkowski valuations is established. Explicit formulas relating previously obtained descriptions of such valuations with the new more accessible one are also derived. Moreover, the action of Alesker's Hard Lefschetz operators on these Minkowski valuations is explored in detail.