Structure of the unitary valuation algebra

Abstract

S. Alesker has shown that if $G$ is a compact subgroup of O(n) acting transitively on the unit sphere $S^{n-1}$ then the vector space $Val^G$ of continuous, translation-invariant, $G$-invariant convex valuations on $R^n$ has the structure of a finite dimensional graded algebra over $R$ satisfying Poincare duality. We show that the kinematic formulas for $G$ are determined by the product pairing. Using this result we then show that the algebra $Val^{U(n) }$ is isomorphic to $R[s,t]/(f_{n+1}, f_{n+2})$, where $s,t$ have degrees 2 and 1 respectively, and the polynomial $f_i$ is the degree $i$ term of the power series $\log(1 + s +t)$.