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Universality Theorems for Inscribed Polytopes and Delaunay Triangulations

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Abstract

We prove that every primary basic semi-algebraic set is homotopy equivalent to the set of inscribed realizations (up to Möbius transformation) of a polytope. If the semi-algebraic set is, moreover, open, it is, additionally, (up to homotopy) the retract of the realization space of some inscribed neighborly (and simplicial) polytope. We also show that all algebraic extensions of \({\mathbb {Q}}\) are needed to coordinatize inscribed polytopes. These statements show that inscribed polytopes exhibit the Mnëv universality phenomenon. Via stereographic projections, these theorems have a direct translation to universality theorems for Delaunay subdivisions. In particular, the realizability problem for Delaunay triangulations is polynomially equivalent to the existential theory of the reals.

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Acknowledgments

K.A. Adiprasito acknowledges the support by an EPDI postdoctoral fellowship and the support by a Minerva Fellowship from the Max Planck Society and by the Romanian NASR, CNCS—UEFISCDI, project PN-II-ID-PCE-2011-3-0533. The research of A. Padrol is supported by the DFG Collaborative Research Center SFB/TR 109 “Discretization in Geometry and Dynamics.” The research of L. Theran was carried out at Inst. Math., Freie Universität Berlin and was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 247029-SDModels. A preliminary version of some of the results of this paper has appeared in [21].

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Authors and Affiliations

  1. Einstein Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel

    Karim A. Adiprasito

  2. Institut für Mathematik, Freie Universität Berlin, Berlin, Germany

    Arnau Padrol

  3. Aalto Science Institute and Department of Computer Science, Aalto University, Aalto, Finland

    Louis Theran

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  1. Karim A. Adiprasito
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  2. Arnau Padrol
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  3. Louis Theran
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Correspondence to Arnau Padrol.

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Adiprasito, K.A., Padrol, A. & Theran, L. Universality Theorems for Inscribed Polytopes and Delaunay Triangulations. Discrete Comput Geom 54, 412–431 (2015). https://doi.org/10.1007/s00454-015-9714-x

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