Non-central Funk-Radon transforms: Single and multiple

We study Funk-type transforms associated with intersections of the unit sphere in Rⁿ with lower-dimensional affine planes passing through a given point inside or outside the sphere. Our goal is to investigate injectivity of such “paired” transforms generated by two families of planes centered at distinct points. Necessary and sufficient conditions of injectivity are obtained in terms of geometry of the centers location. The technique used is based upon the action of the automorphism group of the unit ball and a related billiard-like dynamics on the sphere.