Abstract.
For n≥3 distinct points in the d-dimensional unit sphere 
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.
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Acknowledgments.
I would like to thank Alexander Bobenko and Günter Ziegler for making me familiar with the problem of finding unique representatives for polyhedral types, and Ulrich Pinkall, who has provided the essential insight for this solution.
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Mathematics Subject Classification (2000): 52B10
The author is supported by the DFG Research Center “Mathematics for key technologies”.
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Springborn, B. A unique representation of polyhedral types. Centering via Möbius transformations. Math. Z. 249, 513–517 (2005). https://doi.org/10.1007/s00209-004-0713-5
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DOI: https://doi.org/10.1007/s00209-004-0713-5