Hermitian integral geometry

TL;DR

This paper derives explicit formulas for the affine unitary group's action on complex space, introduces algebraic methods for computing unitary kinematic formulas, and explores invariant valuation bases and their positivity properties.

Contribution

It provides explicit principal kinematic formulas for the affine unitary group and introduces algebraic techniques and canonical bases for unitary-invariant valuations.

Findings

Explicit principal kinematic formula for affine unitary group

Algebraic method for computing unitary kinematic formulas

Characterization of positive and monotone valuation cones

Abstract

We give in explicit form the principal kinematic formula for the action of the affine unitary group on $\C^n$, together with a straightforward algebraic method for computing the full array of unitary kinematic formulas, expressed in terms of certain convex valuations introduced, essentially, by H. Tasaki. We introduce also several other canonical bases for the algebra of unitary-invariant valuations, explore their interrelations, and characterize in these terms the cones of positive and monotone elements.