Discrete & Computational Geometry

, Volume 51, Issue 2, pp 362–393 | Cite as

Empty Monochromatic Simplices

  • Oswin Aichholzer
  • Ruy Fabila-Monroy
  • Thomas Hackl
  • Clemens Huemer
  • Jorge Urrutia
Article

Abstract

Let S be a k-colored (finite) set of n points in \(\mathbb{R}^{d}\), d≥3, in general position, that is, no (d+1) points of S lie in a common (d−1)-dimensional hyperplane. We count the number of empty monochromatic d-simplices determined by S, that is, simplices which have only points from one color class of S as vertices and no points of S in their interior. For 3≤kd we provide a lower bound of \(\varOmega(n^{d-k+1+2^{-d}})\) and strengthen this to Ω(nd−2/3) for k=2.

On the way we provide various results on triangulations of point sets in \(\mathbb{R}^{d}\). In particular, for any constant dimension d≥3, we prove that every set of n points (n sufficiently large), in general position in \(\mathbb{R}^{d}\), admits a triangulation with at least dn+Ω(logn) simplices.

Keywords

Colored point sets Empty monochromatic simplices High dimensional triangulations Simplicial complex 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Oswin Aichholzer
    • 1
  • Ruy Fabila-Monroy
    • 2
  • Thomas Hackl
    • 1
  • Clemens Huemer
    • 3
  • Jorge Urrutia
    • 4
  1. 1.Institute for Software TechnologyUniversity of TechnologyGrazAustria
  2. 2.Departamento de MatemáticasCinvestavD.F. MéxicoMéxico
  3. 3.Departament de Matemàtica Aplicada IVUniversitat Politècnica de CatalunyaBarcelonaSpain
  4. 4.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoD.F. MéxicoMéxico

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