How many cages midscribe an egg?

Abstract

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 $Q\subset {\mathbb{RP}}^3$ combinatorially equivalent to $P$ which midscribes $K$. 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 Teichmüller 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.

15 pages, 1 figure