Centering Koebe polyhedra via M\"obius transformations

TL;DR

This paper explores how various centers of Koebe polyhedra can be achieved through M"obius transformations, extending previous results about barycenters to other polyhedron centers using hyperbolic space vector fields.

Contribution

It generalizes Springborn's result by demonstrating that multiple polyhedron centers can be obtained via M"obius transformations, using topological analysis of vector fields in hyperbolic space.

Findings

Most Koebe polyhedron centers cannot be realized as centers of measures on the sphere.

The proof involves topological properties of integral curves in hyperbolic space.

The results extend the understanding of M"obius transformations in polyhedral geometry.

Abstract

A variant of the Circle Packing Theorem states that the combinatorial class of any convex polyhedron contains elements midscribed to the unit sphere centered at the origin, and that these representatives are unique up to M\"obius transformations of the sphere. Motivated by this result, various papers investigate the problem of centering spherical configurations under M\"obius transformations. In particular, Springborn proved that for any discrete point set on the sphere there is a M\"obius transformation that maps it into a set whose barycenter is the origin, which implies that the combinatorial class of any convex polyhedron contains an element midsribed to a sphere with the additional property that the barycenter of the points of tangency is the center of the sphere. This result was strengthened by Baden, Krane and Kazhdan who showed that the same idea works for any reasonably nice measure defined on the sphere. The aim of the paper is to show that Springborn's statement remains true if we replace the barycenter of the tangency points by many other polyhedron centers. The proof is based on the investigation of the topological properties of the integral curves of certain vector fields defined in hyperbolic space. We also show that most centers of Koebe polyhedra cannot be obtained as the center of a suitable measure defined on the sphere.