Abstract
LetS be a set ofn points in ℝd. A setW is aweak ε-net for (convex ranges of)S if, for anyT⊆S containing εn points, the convex hull ofT intersectsW. We show the existence of weak ε-nets of size\(O((1/\varepsilon ^d )\log ^{\beta _d } (1/\varepsilon ))\), whereβ 2=0,β 3=1, andβ d ≈0.149·2d-1(d-1)!, improving a previous bound of Alonet al. Such a net can be computed effectively. We also consider two special cases: whenS is a planar point set in convex position, we prove the existence of a net of sizeO((1/ε) log1.6(1/ε)). In the case whereS consists of the vertices of a regular polygon, we use an argument from hyperbolic geometry to exhibit an optimal net of sizeO(1/ε), which improves a previous bound of Capoyleas.
Article PDF
Similar content being viewed by others
References
Alon, N., Bárány, I., Füredi, Z., and Kleitman, D. Point selections and weak ε-nets for convex hulls,Combin. Probab. Comput. 3 (1992), 189–200.
Alon, N., and Kleitman, D. Piercing convex sets and the Hadwiger Debrunner (p, q)-problem,Adv. in Math. 96 (1992), 103–112.
Aronov, B., Chazelle, B., Edelsbrunner, H., Guibas, L. J., Sharir, M., and Wenger, R. Points and triangles in the plane and halving planes in space,Discrete Comput. Geom. 6 (1991), 435–442.
Berger, M.Geometry II, Springer-Verlag, New York, 1987.
Capoyleas, V. An almost linear upper bound for weak ε-nets of points in convex position, Manuscript, 1992.
Chazelle, B., Edelsbrunner, H., Guibas, L. J., Hershberger, J., Seidel, R., and Sharir, M. Slimming down by adding; selecting heavily covered points,Proc. 6th ACM Symp. on Computational Geometry, 1990, pp. 116–127.
Coxeter, H. M. S.,Non-Euclidean Geometry, University of Toronto Press, Toronto, 1942.
Edelsbrunner, H.Algorithms in Combinatorial Geometry, Springer-Verlag, Heidelberg, 1987.
Epstein, D. B. A.Analytical and Geometric Aspects of Hyperbolic Space, London Mathematical Society Lecture Notes Series, Vol. 111, Cambridge University Press, Cambridge, 1984.
Fenchel, W.Elementary Geometry in Hyperbolic Space, de Gruyter Studies in Mathematics, Vol. 11, de Gruyter, Berlin, 1989.
Haussler, D., and Welzl, E. Epsilon nets and simplex range queries,Discrete Comput. Geom. 2 (1987), 127–151.
Magnus, W.Noneuclidean Tesselations and Their Groups, Academic Press, New York, 1974.
Matoušek, J. Approximations and optimal geometric divide-and-conquer,Proc. 23rd Annual ACM Symp. on Theory of Computing, 1991, pp. 506–511.
Matoušek, J. Private communication, 1992.
Milnor, J. Hyperbolic geometry: the first 150 years,Bull. Amer. Math. Soc. 6 (1982), 9–24.
Thurston, W. P. Three-dimensional geometry and topology, Preprint, 1993.
Author information
Authors and Affiliations
Additional information
Work by Bernard Chazelle has been supported by NSF Grant CCR-90-02352 and the Geometry Center. Work by Herbert Edelsbrunner has been supported by NSF Grant CCR-89-21421. Work by Michelangelo Grigni has been supported by NSERC Operating Grants and NSF Grant DMS-9206251. Work by Leonidas Guibas and Micha Sharir has been supported by a grant from the U.S.-Israeli Binational Science Foundation. Work by Emo Welzl and Micha Sharir has been supported by a grant from the G.I.F., the German-Israeli Foundation for Scientific Research and Development. Work by Micha Sharir has also been supported by NSF Grant CCR-91-22103, and by a grant from the Fund for Basic Research administered by the Israeli Academy of Sciences.
Rights and permissions
About this article
Cite this article
Chazelle, B., Edelsbrunner, H., Grigni, M. et al. Improved bounds on weak ε-nets for convex sets. Discrete Comput Geom 13, 1–15 (1995). https://doi.org/10.1007/BF02574025
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02574025