Academic Journal
Maximum Betti Numbers of Čech Complexes
| Τίτλος: | Maximum Betti Numbers of Čech Complexes |
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| Συγγραφείς: | Edelsbrunner, Herbert, Pach, János |
| Συνεισφορές: | Herbert Edelsbrunner and János Pach |
| Στοιχεία εκδότη: | Schloss Dagstuhl – Leibniz-Zentrum für Informatik |
| Έτος έκδοσης: | 2024 |
| Συλλογή: | DROPS - Dagstuhl Research Online Publication Server (Schloss Dagstuhl - Leibniz Center for Informatics ) |
| Θεματικοί όροι: | Discrete geometry, computational topology, Čech complexes, Delaunay mosaics, Alpha complexes, Betti numbers, extremal questions |
| Περιγραφή: | 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. |
| Τύπος εγγράφου: | article in journal/newspaper conference object |
| Περιγραφή αρχείου: | application/pdf |
| Γλώσσα: | English |
| Relation: | Is Part Of LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024); https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.53 |
| DOI: | 10.4230/LIPIcs.SoCG.2024.53 |
| Διαθεσιμότητα: | https://doi.org/10.4230/LIPIcs.SoCG.2024.53 https://nbn-resolving.org/urn:nbn:de:0030-drops-199981 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.53 |
| Rights: | https://creativecommons.org/licenses/by/4.0/legalcode |
| Αριθμός Καταχώρησης: | edsbas.F3C28BD2 |
| Βάση Δεδομένων: | BASE |
| DOI: | 10.4230/LIPIcs.SoCG.2024.53 |
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