Flat-shadow characterization of ellipsoids from enclosing hypersurfaces

Establish that if K is a convex body in R^n (n ≥ 3) and S is an embedded hypersurface homeomorphic to the (n−1)-sphere that encloses K, then the condition that every point light source on S produces a flat shadow boundary on K implies that K is an ellipsoid.

Background

Blaschke proved that flat shadow boundaries for all parallel light directions force a convex body to be an ellipsoid. Subsequent work extended this idea to point sources on polyhedral or nearby surfaces. The paper formulates a general conjecture asserting that flatness of shadow boundaries for point sources on any enclosing hypersurface characterizes ellipsoids. The authors confirm the conjecture for convex bodies with sufficiently smooth boundaries, leaving the general non-smooth case open.

References

Motivated by these developments, the following conjecture has emerged as a central open problem: 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. If any point light source on S creates a flat shadow boundary 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 Conjecture 1 (con:01), Section 1 (Introduction)