Improved bounds on weak ε-nets for convex sets
Chazelle, Bernard
Edelsbrunner, Herbert
Grigni, Michelangelo
Guibas, Leonidas
Sharir, Micha
Welzl, Emo
Let S be a set of n points in ℝd . A set W is a weak ε-net for (convex ranges of)S if, for any T⊆S containing εn points, the convex hull of T intersects W. We show the existence of weak ε-nets of size {Mathematical expression}, where β2=0, β3=1, and βd ≈0.149·2d-1(d-1)!, improving a previous bound of Alon et al. Such a net can be computed effectively. We also consider two special cases: when S is a planar point set in convex position, we prove the existence of a net of size O((1/ε) log1.6(1/ε)). In the case where S consists of the vertices of a regular polygon, we use an argument from hyperbolic geometry to exhibit an optimal net of size O(1/ε), which improves a previous bound of Capoyleas.
Springer
1995
info:eu-repo/semantics/article
doc-type:article
text
http://purl.org/coar/resource_type/c_6501
https://research-explorer.app.ist.ac.at/record/4035
Chazelle B, Edelsbrunner H, Grigni M, Guibas L, Sharir M, Welzl E. Improved bounds on weak ε-nets for convex sets. <i>Discrete & Computational Geometry</i>. 1995;13(1):1-15. doi:<a href="https://doi.org/10.1007/BF02574025">10.1007/BF02574025</a>
eng
info:eu-repo/semantics/altIdentifier/doi/10.1007/BF02574025
info:eu-repo/semantics/altIdentifier/issn/0179-5376
info:eu-repo/semantics/closedAccess