How many cages midscribe an egg?

2014/12/15 by Jinsong Liu, Ze Zhou, Liu, Jinsong +1 · 3 citations
Mathematics · #52A15 #52B10 #57Q99 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1412.5430

openalex publication_date 2014/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Midscribability Theorem, which was first proved by O. Schramm, states that: given a strictly convex body K⊂ℝ³ with smooth boundary and a convex polyhedron P, there exists a polyhedron Q ⊂ \mathbbRP³ 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.

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