A Unique Representation of Polyhedral Types. Centering via Möbius Transformations

For n ≥ 3 distinct points in the d-dimensional unit sphere S ⊂ R , there exists a Möbius transformation such that the barycenter of the transformed points is the origin. This Möbius transformation is unique up to post-composition by a rotation. We prove this lemma and apply it to prove the uniqueness part of a representation theorem for 3-dimensional polytopes as claimed by Ziegler (1995): For each polyhedral type there is a unique representative (up to isometry) with edges tangent to the unit sphere such that the origin is the barycenter of the points where the edges touch the sphere. In today’s language, Steinitz’ fundamental theorem of convex types [14], [15] (for a modern treatment see [7], [17]) states that the combinatorial types of convex 3-dimensional polyhedra correspond to the strongly regular cell decompositions of the 2-sphere. (A cell complex is regular if the closed cells are attached without identifications on the boundary. A regular cell complex is strongly regular if the intersection of two closed cells is a closed cell or empty.) Grünbaum and Shephard [8] posed the question whether for every combinatorial type there is a polyhedron with edges tangent to a sphere. This question has been answered affirmatively: Theorem 1 (Koebe [10], Andreev [1][2], Thurston [16], Brightwell and Scheinerman [5], Schramm [12]). For every combinatorial type of convex 3-dimensional polyhedra, there is a representative with edges tangent to the unit sphere. This representative is unique up to projective transformations which fix the sphere and do not make the polyhedron intersect the plane at infinity. A proof which makes use of a variational principle was given by A. Bobenko and the author [4], [13]. The purpose of this article is to prove Theorem 2 below, which singles out a unique representative for each convex type. (The proof given here is also contained in the author’s doctoral dissertation [13].) The claim of Theorem 2 is not new (see Ziegler [17], p. 118, and the second edition of Grünbaum’s classic [7], p. 296a) but this proof seems to be. Theorem 2. For every combinatorial type of convex 3-dimensional polyhedra there is a unique polyhedron (up to isometry) with edges tangent to the unit sphere S ⊂ R , such that the origin 0 ∈ R is the barycenter of the points where the edges touch the sphere. Theorem 2 follows from Theorem 1 and Lemma 1 below (with d = 2). Indeed, the projective transformations of RP that fix S correspond to the Möbius transformations of S. Lemma 1 is also of interest in its own right. Date: 16 March 2004. The author is supported by DFG Research Center Matheon.