Abstract. Let d ≥ 2 and let K ⊂ R d be a convex body containing the origin 0 in its interior. Let, for each direction ω, the (d − 1)-volume of the intersection of K and an arbitrary hyperplane with normal ω attain its maximum if the hyperplane contains 0. Then K is symmetric about 0. The proof uses a linear integro-differential operator on S d−1 , whose null-space needs to be, and will be determined.

In a recent paper the authors have proved that a convex body K R d, d = 2, containing the origin O in its interior, is symmetric with respect to O if and only if V d - 1 (K \ H0) = V d - 1 (K \ H) for all hyperplanes H;H0 such that H and H0 are parallel and H0 3 O (V d - 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 suficiently nice function f on S d - 1 the function R (1) f, where(R (1) f)(.)= R S d - 1 \ . ? (f=)d {with. 2 S d - 1 as pole and as geographic latitude {and have determined the null-space of the operator R (1) . In this paper we extend the definition to any integer m = 1, defining (R (m) f)(.) analogously as for m=1, but using  m f= m rather than f= . (Thecasem=0 is the spherical Radon transformation (Funk transformation).) We investigate the null-space of the operator R (m) : up to a summand of finite dimension, it consists of the even (odd) functions in the domain of the operator, for m odd (even). For the proof we use spherical harmonics.