Year: 2025
Type: article
Abstract: 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 $${{\mathbb R}}^d$$ R d and any radius satisfies $${\beta }_{p}{} = O(N^{m})$$ β p = O ( N m ) , with $$m = \min \{ p+1, {\big \lceil d/2 \big \rceil } \}$$ m = min { p + 1 , ⌈ d / 2 ⌉ } . We construct sets in even and odd dimensions that prove this upper bound is asymptotically tight. For example, we describe a set of $$N = 2(n+1)$$ N = 2 ( n + 1 ) points in $${{\mathbb R}}^3$$ R 3 and two radii such that the first Betti number of the Čech complex at one radius is $$(n+1)^2 - 1$$ ( n + 1 ) 2 - 1 , and the second Betti number of the Čech complex at the other radius is $$n^2$$ n 2 .
Authors: Herbert Edelsbrunner, János Pach
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4416088385