Flat intersections of paired supporting cones imply ellipsoids

Prove that if K is a convex body in R^n (n ≥ 3) and S is an embedded hypersurface enclosing K that is star-convex with respect to the origin, and if for every x in S there exists y in S on the same line through the origin such that the intersection of the boundaries of the supporting cones of K with apexes at x and y is flat, then K is an ellipsoid.

Background

The paper develops a duality between supporting cones and affine sections, linking illumination problems with classical symmetry characterizations of quadrics. Within this framework, a conjecture due to Morales-Amaya asks whether flatness of the intersection of paired supporting cones along lines through the origin forces K to be an ellipsoid. The authors restate this conjecture, relate it to a dual homothety condition on parallel sections, and discuss known partial results.

References

Namely, he posed the following conjecture: Let K⊂ℝn, n≥3, be a convex body and let S be a hypersurface, which is the image of an embedding of the sphere S{n−1}, such that K is contained in the interior of S, and S is star-convex with respect to the origin. Suppose that for every x∈S there exists a different y∈S such that the line xy passes through o and the intersection of the boundaries of supporting cones of K with apexes at x and y is flat. Then K is an ellipsoid.

— On flat shadow boundaries from point light sources and the characterization of ellipsoids  (2603.29130 - Zawalski, 31 Mar 2026) in Conjecture 3 (con:03), Section 2.4 (Flat intersections of supporting cones)