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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

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Volume 16, Issue 3

Issues

Neighborly inscribed polytopes and delaunay triangulations

Bernd Gonska / Arnau Padrol
  • Corresponding author
  • Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Pierre et Marie Curie, Case 247, 4 Place Jussieu, 75252 Paris Cedex 05, France
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Published Online: 2016-06-27 | DOI: https://doi.org/10.1515/advgeom-2015-0045

Abstract

We construct a large family of neighborly polytopes that can be realized with all the vertices on the boundary of any smooth strictly convex body. In particular, we show that for d ≥ 4 there are superexponentially many combinatorially distinct neighborly d-polytopes on n vertices that admit realizations inscribed in the sphere. These are the first examples of inscribable neighborly polytopes that are not cyclic polytopes, and provide the current best lower bound for the number of combinatorial types of inscribable polytopes (which coincides with the current best lower bound for the number of combinatorial types of polytopes). Via stereographic projections, this translates into a superexponential lower bound for the number of combinatorial types of (neighborly) Delaunay triangulations in ℝd for d ≥ 3.

Keywords: Neighborly polytope; inscribed polytope; Delaunay triangulation; number of combinatorial types

Communicated by: M. Henk

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About the article

Received: 2014-04-24

Revised: 2015-02-20

Published Online: 2016-06-27

Published in Print: 2016-07-01


Citation Information: Advances in Geometry, Volume 16, Issue 3, Pages 349–360, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2015-0045.

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