Unitarily invariant valuations and Tutte's sequence

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.

The module of unitarily invariant area measures

The hermitian analog of Aleksandrov's area measures of convex bodies is investigated. A characterization of those area measures which arise as the first variation of unitarily invariant valuations is established. General smooth area measures are shown to form a module over smooth valuations and the module of unitarily invariant area measures is described explicitly.

凸函数上酉不变估值的几何分解

A geometric decomposition for unitarily invariant valuations on convex functions

本文对 $\mathbb{C}^n$ 上凸函数空间中连续、对偶外延平移不变且 $\mathrm{U}(n)$-不变的估值进行了完全分类，证明其空间可按特殊子空间上的消失性质分解为子空间直和，并给出基于两族 Monge-Ampère 型算子主值积分的唯一表示。

Dual curvature measures in hermitian integral geometry

The local kinematic formulas on complex space forms induce the structure of a commutative algebra on the space $\mathrm{Curv}^{\mathrm{U}(n)*}$ of dual unitarily invariant curvature measures. Building on the recent results from integral geometry in complex space forms, we describe this algebra structure explicitly as a polynomial algebra. This is a short way to encode all local kinematic formulas. We then characterize the invariant valuations on complex space forms leaving the space of invariant angular curvature measures fixed.