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 Sn-1, then the vector space ValG 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 ValU(n) is isomorphic to R[s, t]/(fn+1, fn+2), where s, t have degrees 2 and 1 respectively, and the polynomial fi is the degree i term of the power series log(1 + s + t).