Overdetermined transforms in integral geometry

We consider an example of the Gelfand-Gindikin n-dimensional complex of k-dimensional planes in Rⁿ. Sharp existence conditions and inversion formulas are obtained for the corresponding restricted k-plane transform of L^p functions. Similar results are obtained for overdetermined Radon type transforms on the sphere and the hyperbolic space. A topological isomor-phism of the relevant Schwartz spaces with respect to the restricted k-plane transform is established. Related open problems, in particular, the restricted lower dimensional Busemann-Petty problem for sections of convex bodies, are discussed.