# Dual area measures and local additive kinematic formulas

## TL;DR

This paper establishes new structural results for area measures and introduces a convolution product on dual area measures, enabling explicit derivation of local additive kinematic formulas in hermitian vector spaces.

## Contribution

It introduces the space of smooth dual area measures and a convolution product that encodes local additive kinematic formulas for transitive group actions.

## Key findings

- Higher moment maps are injective on area measures.
- Kernel of the centroid map equals the image of the first variation map.
- Explicit local additive kinematic formulas in hermitian vector spaces.

## Abstract

We prove that higher moment maps on area measures of a euclidean vector space are injective, while the kernel of the centroid map equals the image of the first variation map. Based on this, we introduce the space of smooth dual area measures on a finite-dimensional euclidean vector space and prove that it admits a natural convolution product which encodes the local additive kinematic formulas for groups acting transitively on the unit sphere. As an application of this new integral-geometric structure, we obtain the local additive kinematic formulas in hermitian vector spaces in a very explicit way.