Two-body shadow-flatness conjecture for concentric homothetic ellipsoids

Establish that if K and L are convex bodies in R^n (n ≥ 3) with L ⊆ K, and for every boundary point p of K there exist at least L(n) point light sources on the supporting cone of L with apex p such that L casts a shadow with flat boundary on K, then K and L must be concentric, homothetic ellipsoids.

Background

The authors show how their main theorem implies a recent result characterizing pairs of concentric homothetic ellipsoids under reciprocal flat-shadow conditions. They then propose a stronger, more flexible formulation involving light sources on supporting cones of L that would subsume both the main theorem and existing two-body results if true.