We show that a centrally symmetric convex body is determined, up to translation, by any Radon transform of one of its projection functions. It is also proved that all Radon transforms are injective when restricted to projection functions of polytopes. Both results were known previously in the case of Radon transforms Ri,j with i<j. Here we use harmonic analysis on Grassmann manifolds to show that these results extend to all Radon transforms.