How many cages midscribe an egg?

The Midscribability Theorem, which was first proved by O. Schramm, states that: given a strictly convex body $K\subset {\mathbb{R}}^3$ with smooth boundary and a convex polyhedron $P$, there exists a polyhedron $Q \subset $\mathbb{R}\mathbb{P}$^3$combinatoriallyequivalentto$P$whichmidscribes$K$. Here the word “midscribe” means that all it’s edges are tangent to the boundary surface of $K$. By using of the intersection number technique, together with the Teichm"{u}ller theory of packings, this paper provides an alternative approach to this theorem. Furthermore, combining Schramm’s method with the above ones, the authors prove a rigidity result concerning this theorem as well. Namely, such a polyhedron is unique under certain normalization conditions.