On an Integro-differential Transform on the Sphere

Abstract. In a recent paper the authors have proved that a convex body K ⊂ R, d ≥ 2, containing the origin 0 in its interior, is symmetric with respect to 0 if and only if Vd−1(K ∩ H ) ≥ Vd−1(K ∩ H) for all hyperplanes H,H ′ such that H and H are parallel and H ∋ 0 (Vd−1 is (d − 1)–measure). For the proof the authors have employed a new type of integro–differential transform, that lets to correspond to a sufficiently nice function f on Sd−1 the function R(1)f , where (R(1)f)(ξ) =