Integral geometry of unitary area measures

TL;DR

This paper establishes the existence of kinematic formulas for unitary area measures, describing their algebraic structure and providing an explicit algorithm for deriving these formulas.

Contribution

It introduces a new algebraic framework for unitary area measures and provides an explicit algorithm for kinematic formulas, extending classical integral geometry results.

Findings

Kinematic operator has a co-product structure.

Explicit algebraic description for the unitary case.

Algorithm for deriving kinematic formulas.

Abstract

The existence of kinematic formulas for area measures with respect to any connected, closed subgroup of the orthogonal group acting transitively on the unit sphere is established. In particular, the kinematic operator for area measures is shown to have the structure of a co-product. In the case of the unitary group the algebra associated to this co-product is described explicitly in terms of generators and relations. As a consequence, a simple algorithm that yields explicit kinematic formulas for unitary area measures is obtained.