On spheres with k points inside LIPIcs Edelsbrunner, Herbert ; https://orcid.org/0000-0002-9823-6833 Garber, Alexey Saghafian, Morteza ddc:510 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. Schloss Dagstuhl - Leibniz-Zentrum für Informatik 2025 info:eu-repo/semantics/conferenceObject doc-type:conferenceObject text http://purl.org/coar/resource_type/c_5794 https://research-explorer.ista.ac.at/record/20005 https://research-explorer.ista.ac.at/download/20005/20016 Edelsbrunner H, Garber A, Saghafian M. On spheres with k points inside. In: <i>41st International Symposium on Computational Geometry</i>. Vol 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2025. doi:<a href="https://doi.org/10.4230/LIPIcs.SoCG.2025.43">10.4230/LIPIcs.SoCG.2025.43</a> eng info:eu-repo/semantics/altIdentifier/doi/10.4230/LIPIcs.SoCG.2025.43 info:eu-repo/semantics/altIdentifier/e-issn/1868-8969 info:eu-repo/semantics/altIdentifier/isbn/9783959773706 info:eu-repo/semantics/altIdentifier/arxiv/2410.21204 https://creativecommons.org/licenses/by/4.0/ info:eu-repo/semantics/openAccess