A Hadwiger-type theorem for the special unitary group

TL;DR

This paper computes the dimension of the space of SU(n)-invariant valuations on complex n-space, constructs a geometric basis, and derives kinematic formulas, extending Hadwiger's theorem to the special unitary group.

Contribution

It provides an explicit dimension count, constructs a geometric basis, and derives kinematic formulas for SU(n)-invariant valuations, extending classical valuation theory.

Findings

Dimension formulas for even and odd n

Explicit geometric basis constructed

Kinematic formulas derived for SU(n)

Abstract

The dimension of the space of SU(n) and translation invariant continuous valuations on $\mathbb{C}^n, n \geq 2$ is computed. For even $n$, this dimension equals $(n^2+3n+10)/2$; for odd $n$ it equals $(n^2+3n+6)/2$. An explicit geometric basis of this space is constructed. The kinematic formulas for SU(n) are obtained as corollaries.