  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 $\log(1 + s +t)$.
