Minimal number of local light sources forcing an ellipsoid

Prove that for a convex body K in R^n (n ≥ 3) with boundary of class C^3, if at every boundary point p there are at least n+1 point light sources lying on the tangent hyperplane T_p∂K in general linear position with respect to p that generate flat shadow boundaries on K, then K is an ellipsoid.

Background

The main theorems of the paper establish the ellipsoid conclusion under a finite number of flat-shadow conditions with a dimension-dependent constant L(n). The authors provide upper bounds on L(n) using combinatorial arguments and show that n sources do not suffice via ℓp balls. Motivated by evidence and structure of their proofs, they conjecture the sharp threshold is n+1 sources per boundary point.

References

Keeping all this in mind, and on the other hand, having a counterexample showing that n light sources are not enough, the following conjecture seems promising to us: Let K⊂ℝn, n≥3, be a convex body with boundary of class C3. Suppose that for every point p∈∂K on the boundary, there are at least n+1 point light sources on the tangent hyperplane T_p∂K in a general linear position with respect to p, that create flat shadow boundaries on K. 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 Section 6.2 (Lower bound)