conference paper published yes Herbert Edelsbrunner author 3FB178DA-F248-11E8-B48F-1D18A9856A870000-0002-9823-6833 Alexey Garber author Morteza Saghafian author f86f7148-b140-11ec-9577-95435b8df824 HeEd department SoCG: Symposium on Computational Geometry Mathematics, Computer Science project Persistence and stability of geometric complexes project 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. https://research-explorer.ista.ac.at/download/20005/20016/2025_LIPIcs.SoCG_Edelsbrunner.pdf application/pdfno Schloss Dagstuhl - Leibniz-Zentrum für Informatik2025Kanazawa, Japan eng 1868-8969 9783959773706 2410.2120410.4230/LIPIcs.SoCG.2025.43 332 Edelsbrunner, H., Garber, A., &#38; Saghafian, M. (2025). On spheres with k points inside. In <i>41st International Symposium on Computational Geometry</i> (Vol. 332). Kanazawa, Japan: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href="https://doi.org/10.4230/LIPIcs.SoCG.2025.43">https://doi.org/10.4230/LIPIcs.SoCG.2025.43</a> H. Edelsbrunner, A. Garber, and M. Saghafian, “On spheres with k points inside,” in <i>41st International Symposium on Computational Geometry</i>, Kanazawa, Japan, 2025, vol. 332. 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> Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “On Spheres with k Points Inside.” In <i>41st International Symposium on Computational Geometry</i>, Vol. 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. <a href="https://doi.org/10.4230/LIPIcs.SoCG.2025.43">https://doi.org/10.4230/LIPIcs.SoCG.2025.43</a>. H. Edelsbrunner, A. Garber, M. Saghafian, in:, 41st International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. Edelsbrunner H, Garber A, Saghafian M. 2025. On spheres with k points inside. 41st International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 332, 43. Edelsbrunner, Herbert, et al. “On Spheres with k Points Inside.” <i>41st International Symposium on Computational Geometry</i>, vol. 332, 43, 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>. 200052025-07-13T22:01:22Z2025-07-14T07:26:14Z