Tensor valuations

本计画旨在深入研究各类集合之张量泛函之积分几何结果与形态性质。一个主要的重点将是translative，运动学和旋转积分几何公式张量估值和不一定添加剂张量泛函。我们将介绍相当一般的集合类和不同的生成测度的张量值泛函。这将使我们能够将张量估值作为与各向异性微结构材料中的物理过程相关的形态学度量，并用于非球形粒子系综的热力学。张量泛函将在该提案的所有项目中作为几何描述符发挥重要作用。本项目的一个特别重点将是基础问题。

This project aims at a thorough study of integral geometric results and morphological properties of tensorial functionals of various classes of sets. One main focus will be on translative, kinematic and rotational integral geometric formulas for tensor valuations and for not necessarily additive tensor functionals. We will introduce tensor-valued functionals for quite general classes of sets and with respect to different generating measures. This will enable us to apply tensor valuations as morphological measures that are relevant to the physical processes in anisotropic microstructured materials, and for the thermodynamics of ensembles of non-spherical particles. Tensor functionals will play an important role as geometric descriptors in all projects of the proposal. A particular emphasis in this project will be on foundational problems.