A characterization of centrally symmetric convex bodies in terms of visual cones

Abstract We prove the following result: Let K be a strictly convex body in the Euclidean space ℝn, n ≥ 3, and let L be a hypersurface which is the image of an embedding of the sphere 𝕊n–1, such that K is contained in the interior of L. Suppose that, for every x ∈ L, there exists y ∈ L such that the support cones of K with apexes at x and y differ by a central symmetry. Then K and L are centrally symmetric and concentric.