Minkowski Valuations and Generalized Valuations

TL;DR

This paper introduces a convolution-based framework for Minkowski valuations, classifies generalized spherical valuations, and derives Crofton and kinematic formulas, advancing the mathematical understanding of valuation theory.

Contribution

It provides a new convolution representation and classification of Minkowski valuations, leading to novel Crofton and kinematic formulas.

Findings

Convolution representation of Minkowski valuations established

Classification of generalized spherical valuations achieved

Crofton and kinematic formulas derived for Minkowski valuations

Abstract

A convolution representation of continuous translation invariant and SO(n) equivariant Minkowski valuations is established. This is based on a new classification of translation invariant generalized spherical valuations. As applications, Crofton and kinematic formulas for Minkowski valuations are obtained.