conference paper published yes Herbert Edelsbrunner author 3FB178DA-F248-11E8-B48F-1D18A9856A870000-0002-9823-6833 János Pach author E62E3130-B088-11EA-B919-BF823C25FEA4 HeEd department SoCG: Symposium on Computational Geometry Alpha Shape Theory Extended project Persistence and stability of geometric complexes project Mathematics, Computer Science project The Upper Bound Theorem for convex polytopes implies that the p-th Betti number of the Čech complex of any set of N points in ℝ^d and any radius satisfies β_p = O(N^m), with m = min{p+1, ⌈d/2⌉}. We construct sets in even and odd dimensions, which prove that this upper bound is asymptotically tight. For example, we describe a set of N = 2(n+1) points in ℝ³ and two radii such that the first Betti number of the Čech complex at one radius is (n+1)² - 1, and the second Betti number of the Čech complex at the other radius is n². In particular, there is an arrangement of n contruent balls in ℝ³ that enclose a quadratic number of voids, which answers a long-standing open question in computational geometry. https://research-explorer.ista.ac.at/download/17146/17152/2024_LIPICS_Edelsbrunner.pdf application/pdfno Schloss Dagstuhl - Leibniz-Zentrum für Informatik2024Athens, Greece eng 1868-8969 9783959773164 2310.1480110.4230/LIPIcs.SoCG.2024.53 293 https://research-explorer.ista.ac.at/record/20657 Edelsbrunner H, Pach J. 2024. Maximum Betti numbers of Čech complexes. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 53. Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href="https://doi.org/10.4230/LIPIcs.SoCG.2024.53">10.4230/LIPIcs.SoCG.2024.53</a> Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” <i>40th International Symposium on Computational Geometry</i>, vol. 293, 53, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href="https://doi.org/10.4230/LIPIcs.SoCG.2024.53">10.4230/LIPIcs.SoCG.2024.53</a>. H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293. Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” In <i>40th International Symposium on Computational Geometry</i>, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href="https://doi.org/10.4230/LIPIcs.SoCG.2024.53">https://doi.org/10.4230/LIPIcs.SoCG.2024.53</a>. Edelsbrunner, H., &#38; Pach, J. (2024). Maximum Betti numbers of Čech complexes. In <i>40th International Symposium on Computational Geometry</i> (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href="https://doi.org/10.4230/LIPIcs.SoCG.2024.53">https://doi.org/10.4230/LIPIcs.SoCG.2024.53</a> H. Edelsbrunner, J. Pach, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. 171462024-06-16T22:01:06Z2025-12-01T15:19:20Z