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 Rⁿ has the structure of a finite dimensional graded algebra over R satisfying Poincaré 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).