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On Spheres with k Points Inside

Authors Herbert Edelsbrunner , Alexey Garber , Morteza Saghafian

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  • Theory of computation → Computational geometry
Keywords
  • Triangulations
  • higher-order Delaunay triangulations
  • hypertriangulations
  • Delone sets
  • k-sets
  • Worpitzky’s identity
  • hypersimplices

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Abstract

We generalize a classical result by Boris Delaunay that introduced Delaunay triangulations. In particular, we prove that for a locally finite and coarsely dense generic point set A in ℝ^d, every generic point of ℝ^d belongs to exactly binom(d+k,d) simplices whose vertices belong to A and whose circumspheres enclose exactly k points of A. We extend this result to the cases in which the points are weighted, and when A contains only finitely many points in ℝ^d or in 𝕊^d. Furthermore, we use the result to give a new geometric proof for the fact that volumes of hypersimplices are Eulerian numbers.

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Herbert Edelsbrunner, Alexey Garber, and Morteza Saghafian. On Spheres with k Points Inside. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 43:1-43:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.43

Author Details

Herbert Edelsbrunner
  • IST Austria (Institute of Science and Technology Austria), Klosterneuburg, Austria
Alexey Garber
  • School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Brownsville, TX, USA
Morteza Saghafian
  • IST Austria (Institute of Science and Technology Austria), Klosterneuburg, Austria

Funding

  • Edelsbrunner, Herbert: partially supported by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35.
  • Garber, Alexey: partially supported by the Simons Foundation.
  • Saghafian, Morteza: partially supported by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35.

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