Local properties of intertwining operators on the sphere

Blaschkeʼs original question regarding the local determination of zonoids (or projection bodies) has been the subject of much research over the years. In recent times this research has been extended to include intersection bodies and it has been shown that neither zonoids nor intersection bodies have local characterizations. However, it has also been proved that both these classes of bodies admit equatorial characterizations in odd dimensions, but not in even dimensions. The proofs of these results were mostly analytic using properties of associated spherical integral transforms, the Cosine transform and the Radon transform.Here we elaborate a general principle, showing that such local or equatorial characterization problems are equivalent to corresponding support properties of the spherical operators. We discuss this within a general framework, for intertwining operators on C∞-functions, and apply the results to further geometric constructions, namely to certain mean section bodies, to Lq-centroid bodies and to k-intersection bodies.