@inproceedings{20005,
  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.},
  author       = {Edelsbrunner, Herbert and Garber, Alexey and Saghafian, Morteza},
  booktitle    = {41st International Symposium on Computational Geometry},
  isbn         = {9783959773706},
  issn         = {1868-8969},
  location     = {Kanazawa, Japan},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{On spheres with k points inside}},
  doi          = {10.4230/LIPIcs.SoCG.2025.43},
  volume       = {332},
  year         = {2025},
}