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We study Funk-type transforms on the unit sphere in R^n associated withcross-sections of the sphere by lower-dimensional planes passing through anarbitrary fixed point inside the sphere or outside. Our main concern isinjectivity of the corresponding paired transforms generated by two families ofplanes centered at distinct points. Necessary and sufficient conditions for thepaired transforms to be injective are obtained, depending on geometricalconfiguration of the centers. Our method relies on the action of theautomorphism group of the unit ball and the relevant billiard-like dynamics onthe sphere.

Injectivity of pairs of non-central Funk transforms