TY - JOUR
AB - 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.
AU - Chazelle, Bernard
AU - Edelsbrunner, Herbert
AU - Grigni, Michelangelo
AU - Guibas, Leonidas
AU - Sharir, Micha
AU - Welzl, Emo
ID - 4035
IS - 1
JF - Discrete & Computational Geometry
SN - 0179-5376
TI - Improved bounds on weak ε-nets for convex sets
VL - 13
ER -