A unique representation of polyhedral types. Centering via Möbius transformations (Q1771986)

The combinatorial types of convex polyhedra in three-dimensional Euclidean space \(R^3\) correspond to the strongly regular cell decompositions of the two-dimensional unit sphere \(S^2\). Moreover, every combinatorial type of convex polyhedra in \(R^3\) may be represented (uniquely up to isometry of \(R^3\)) by a convex polyhedron with edges tangent to \(S^2\), such that the origin \(O\in R^3\) is the barycenter of the points where the edges touch the sphere [see \textit{G. Ziegler}, Lectures on polytopes. Graduate Texts in Mathematics 152 (Berlin: Springer-Verlag) (1995; Zbl 0823.52002), \textit{B. Grünbaum}, Convex polytopes. Graduate Texts in Mathematics 221 (New York, NY: Springer) (2003; Zbl 1033.52001), \textit{P. Koebe}, ``Kontaktprobleme der konformen Abbildung'', Ber. Verh. Sächs. Akad. Leipzig 88, 141--164 (1936; Zbl 0017.21701), \textit{W. P. Thurston}, Three-dimensional geometry and topology. Vol. 1. Ed. by Silvio Levy. Princeton Mathematical Series 35 (Princeton, NJ: Princeton University Press) (1997; Zbl 0873.57001), \textit{O. Schramm}, Invent. Math. 107, No. 3, 543--560 (1992; Zbl 0726.52003), \textit{G. R. Brightwell} and \textit{E. R. Scheinerman}, ``Representations of planar graphs''. SIAM J. Discrete Math. 6, No. 2, 214--229 (1993; Zbl 0782.05026)]. The author proposes a new proof for this statement which is based on the following original lemma: For any set of \(n\geq 3\) points \(v_1,\dots, v_n\) on the \(d\)-dimensional sphere \(S^d\subset R^{d+1}\) there exists a Möbius transformation \(T\) of \(S^d\) such that \(\sum_{j=1}^n Tv_j = 0\).