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As proved by Radon [16] and John [13], a differentiable function / of compact support on a Euclidean space R n can be determined explicitly by means of its integrals over the hyperplanes in the space. Let J(~o, p) denote the integral of / over the hyperplane <x, co> = p where r is a unit vector and <,> the inner product in R n. I f A denotes the Laplacian on R n, do) the area element on the unit sphere S n-1 then (John [14], p. 13)

The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds