On flat shadow boundaries from point light sources and the characterization of ellipsoids

TL;DR

This paper confirms that convex bodies with flat shadow boundaries from point light sources on a hypersurface are ellipsoids, extending classical results and linking illumination properties to symmetry characterizations of quadrics.

Contribution

It proves a conjecture relating point light source illumination to ellipsoid characterization for smooth convex bodies and develops a duality framework connecting illumination and symmetry properties.

Findings

Convex bodies with flat shadow boundaries from point sources are ellipsoids.

Established a duality linking illumination by point sources to hyperplane section symmetries.

Connected classical and new characterizations of quadrics through this framework.

Abstract

In his classical work, W. Blaschke proved that a convex body whose shadow boundaries are flat for every direction of parallel illumination must be an ellipsoid. An extension recently proposed by I. Gonzalez-Garc\'ia, J. Jer\'onimo-Castro, E. Morales-Amaya, and D.J. Verdusco-Hern\'andez predicts that the same conclusion holds for illumination by point light sources located on a hypersurface enclosing the body. We confirm this conjecture for convex bodies with sufficiently smooth boundaries. We further develop a duality framework relating illumination by point light sources to classical symmetry properties of hyperplane sections, linking several known and conjectured characterizations of quadrics from these complementary viewpoints.