eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
53:1
53:14
10.4230/LIPIcs.SoCG.2024.53
article
Maximum Betti Numbers of Čech Complexes
Edelsbrunner, Herbert
1
https://orcid.org/0000-0002-9823-6833
Pach, János
2
1
https://orcid.org/0000-0002-2389-2035
Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
Rényi Institute of Mathematics, Budapest, Hungary
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://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.53/LIPIcs.SoCG.2024.53.pdf
Discrete geometry
computational topology
Čech complexes
Delaunay mosaics
Alpha complexes
Betti numbers
extremal questions