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