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

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.

凸集上赋值的非阿基米德类比的 Hodge-Riemann 关系

Towards Hodge-Riemann relations for non-Archimedean analogs of valuations on convex sets

本文在非阿基米德类比空间中引入乘积与卷积两种乘法结构，提出混合硬 Lefschetz 定理和 Hodge-Riemann 关系的猜想，并证明乘积的度 1 与卷积的余度 1 的非混合 Hodge-Riemann 关系。

From harmonic analysis of translation-invariant valuations to geometric inequalities for convex bodies

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 v…

On mixed Hodge-Riemann relations for translation-invariant valuations and Aleksandrov-Fenchel inequalities

A version of the Hodge-Riemann relations for valuations was recently conjectured and proved in several special cases by the first-named author. The Lefschetz operator considered there arises as either the product or the convolution with the mixed volume of several Euclidean balls. Here we prove that in (co-)degree one the Hodge-Riemann relations persist if the balls are replaced by several different (centrally symmetric) convex bodies with smooth boundary with positive Gauss curvature. While these mixed Hodge-Riemann relations for the convolution directly imply the Aleksandrov-Fenchel inequali…

A Fourier type transform on translation invariant valuations on convex sets

Let $V$ be a finite dimensional real vector space. In the article we construct an isomorphism between the space of smooth translation invariant valuations on convex subsets of $V$ and the space of such valuations (twisted by densities) on the dual space $V^*$. This isomorphism commutes with the natural action of the full linear group on both spaces of valuations, maps product on the source (introduced in arXiv:math/0301148) to the convolution on the target (introduced in arXiv:math/0607449), and satisfies a Plancherel type inversion formula. As an application, a version of the hard Lefschetz t…