Mathematische Zeitschrift

, Volume 249, Issue 3, pp 513–517 | Cite as

A unique representation of polyhedral types. Centering via Möbius transformations

Article

Abstract.

For n≥3 distinct points in the d-dimensional unit sphere Open image in new window 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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© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Technische Universität BerlinInstitut für MathematikBerlinGermany

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