Integral geometry under $G_2$ and $Spin(7)$

TL;DR

This paper establishes a Hadwiger-type theorem for the exceptional Lie groups G_2 and Spin(7), characterizing their invariant valuations, constructing bases, and deriving kinematic formulas.

Contribution

It provides the first complete description of invariant valuations and kinematic formulas for G_2 and Spin(7), including algebra structures and bases.

Findings

Invariant valuation algebras are 10-dimensional for both groups.

Constructed geometrically meaningful bases for these algebras.

Derived explicit kinematic formulas for G_2 and Spin(7).

Abstract

A Hadwiger-type theorem for the exceptional Lie groups $G_2$ and $Spin(7)$ is proved. The algebras of $G_2$ or $Spin(7)$ invariant, translation invariant continuous valuations are both of dimension 10. Geometrically meaningful bases are constructed and the algebra structures are computed. Finally, the kinematic formulas for these groups are determined.