On the Range of the Radon d-Plane Transform and Its Dual

We present direct, group-theoretic proofs of the range theorem for the Radon d-plane transform f → f̂ on Y(Rn). (The original proof, by Richter, involves extensive use of local coordinate calculations on G(d, n), the Grassmann manifold of affine d-planes in Rn.) We show that moment conditions are not sufficient to describe this range when$d < n - 1$, in contrast to the compactly supported case. Finally, we show that the dual d-plane transform maps E(G(d, n)) surjectively onto E(Rn).