Centering Koebe polyhedra via Möbius transformations

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