How many cages midscribe an egg?

TL;DR

This paper offers an alternative proof and a rigidity result for the Midscribability Theorem, which states that a convex polyhedron can be inscribed in a convex body with all edges tangent to its surface.

Contribution

It introduces a new approach using intersection numbers and Teichmüller theory, and proves the uniqueness of the midscribing polyhedron under normalization.

Findings

Provided an alternative proof of the Midscribability Theorem

Established a rigidity result ensuring the uniqueness of the inscribed polyhedron

Combined different methods to deepen understanding of polyhedral inscribability

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 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.