{"publisher":"Springer","article_type":"original","date_updated":"2022-06-13T12:37:06Z","publication":"Discrete & Computational Geometry","day":"01","status":"public","volume":13,"year":"1995","oa_version":"None","date_published":"1995-12-01T00:00:00Z","abstract":[{"lang":"eng","text":"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."}],"publist_id":"2094","citation":{"ama":"Chazelle B, Edelsbrunner H, Grigni M, Guibas L, Sharir M, Welzl E. Improved bounds on weak ε-nets for convex sets. <i>Discrete &#38; Computational Geometry</i>. 1995;13(1):1-15. doi:<a href=\"https://doi.org/10.1007/BF02574025\">10.1007/BF02574025</a>","ieee":"B. Chazelle, H. Edelsbrunner, M. Grigni, L. Guibas, M. Sharir, and E. Welzl, “Improved bounds on weak ε-nets for convex sets,” <i>Discrete &#38; Computational Geometry</i>, vol. 13, no. 1. Springer, pp. 1–15, 1995.","ista":"Chazelle B, Edelsbrunner H, Grigni M, Guibas L, Sharir M, Welzl E. 1995. Improved bounds on weak ε-nets for convex sets. Discrete &#38; Computational Geometry. 13(1), 1–15.","short":"B. Chazelle, H. Edelsbrunner, M. Grigni, L. Guibas, M. Sharir, E. Welzl, Discrete &#38; Computational Geometry 13 (1995) 1–15.","mla":"Chazelle, Bernard, et al. “Improved Bounds on Weak ε-Nets for Convex Sets.” <i>Discrete &#38; Computational Geometry</i>, vol. 13, no. 1, Springer, 1995, pp. 1–15, doi:<a href=\"https://doi.org/10.1007/BF02574025\">10.1007/BF02574025</a>.","apa":"Chazelle, B., Edelsbrunner, H., Grigni, M., Guibas, L., Sharir, M., &#38; Welzl, E. (1995). Improved bounds on weak ε-nets for convex sets. <i>Discrete &#38; Computational Geometry</i>. Springer. <a href=\"https://doi.org/10.1007/BF02574025\">https://doi.org/10.1007/BF02574025</a>","chicago":"Chazelle, Bernard, Herbert Edelsbrunner, Michelangelo Grigni, Leonidas Guibas, Micha Sharir, and Emo Welzl. “Improved Bounds on Weak ε-Nets for Convex Sets.” <i>Discrete &#38; Computational Geometry</i>. Springer, 1995. <a href=\"https://doi.org/10.1007/BF02574025\">https://doi.org/10.1007/BF02574025</a>."},"language":[{"iso":"eng"}],"type":"journal_article","date_created":"2018-12-11T12:06:33Z","issue":"1","title":"Improved bounds on weak ε-nets for convex sets","user_id":"ea97e931-d5af-11eb-85d4-e6957dddbf17","month":"12","publication_status":"published","main_file_link":[{"url":"https://link.springer.com/article/10.1007/BF02574025"}],"doi":"10.1007/BF02574025","article_processing_charge":"No","page":"1 - 15","quality_controlled":"1","acknowledgement":"The authors wish to express their gratitude for the support and hospitality of the DEC Palo Alto Systems Research Center.","intvolume":"        13","extern":"1","author":[{"full_name":"Chazelle, Bernard","last_name":"Chazelle","first_name":"Bernard"},{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"full_name":"Grigni, Michelangelo","last_name":"Grigni","first_name":"Michelangelo"},{"last_name":"Guibas","full_name":"Guibas, Leonidas","first_name":"Leonidas"},{"full_name":"Sharir, Micha","last_name":"Sharir","first_name":"Micha"},{"first_name":"Emo","full_name":"Welzl, Emo","last_name":"Welzl"}],"publication_identifier":{"issn":["0179-5376"]},"_id":"4035"}