TY - JOUR

T1 - Improved bounds on weak ε-nets for convex sets

AU - Chazelle, B.

AU - Edelsbrunner, H.

AU - Grigni, M.

AU - Guibas, L.

AU - Sharir, M.

AU - Welzl, E.

PY - 1995/12

Y1 - 1995/12

N2 - Let S be a set of n points in ℝ d . 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 β 2=0, β 3=1, and β d ≈0.149·2 d-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.

AB - Let S be a set of n points in ℝ d . 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 β 2=0, β 3=1, and β d ≈0.149·2 d-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.

UR - http://www.scopus.com/inward/record.url?scp=51249162757&partnerID=8YFLogxK

U2 - 10.1007/BF02574025

DO - 10.1007/BF02574025

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AN - SCOPUS:51249162757

SN - 0179-5376

VL - 13

SP - 1

EP - 15

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 1

ER -