{"abstract":[{"lang":"eng","text":"A finite set N ⊂ Rd is a weak ε-net for an n-point set X ⊂ Rd (with respect to convex sets) if N intersects every convex set K with |K ∩ X| ≥ εn. We give an alternative, and arguably simpler, proof of the fact, first shown by Chazelle et al., that every point set X in Rd admits a weak ε-net of cardinality O(ε-dpolylog(1/ε)). Moreover, for a number of special point sets (e.g., for points on the moment curve), our method gives substantially better bounds. The construction yields an algorithm to construct such weak ε-nets in time O(n ln(1/ε))."}],"publication_status":"published","publication":"Discrete & Computational Geometry","issue":"2","title":"New constructions of weak ε-nets","quality_controlled":0,"author":[{"last_name":"Matoušek","full_name":"Matoušek, Jiří","first_name":"Jiří"},{"orcid":"0000-0002-1494-0568","full_name":"Uli Wagner","first_name":"Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","last_name":"Wagner"}],"doi":"10.1007/s00454-004-1116-4","_id":"2425","intvolume":"        32","type":"journal_article","extern":1,"volume":32,"date_updated":"2021-01-12T06:57:24Z","publist_id":"4500","date_published":"2004-07-01T00:00:00Z","day":"01","page":"195 - 206","year":"2004","status":"public","publisher":"Springer","date_created":"2018-12-11T11:57:35Z","month":"07","citation":{"short":"J. Matoušek, U. Wagner, Discrete &#38; Computational Geometry 32 (2004) 195–206.","ama":"Matoušek J, Wagner U. New constructions of weak ε-nets. <i>Discrete &#38; Computational Geometry</i>. 2004;32(2):195-206. doi:<a href=\"https://doi.org/10.1007/s00454-004-1116-4\">10.1007/s00454-004-1116-4</a>","mla":"Matoušek, Jiří, and Uli Wagner. “New Constructions of Weak ε-Nets.” <i>Discrete &#38; Computational Geometry</i>, vol. 32, no. 2, Springer, 2004, pp. 195–206, doi:<a href=\"https://doi.org/10.1007/s00454-004-1116-4\">10.1007/s00454-004-1116-4</a>.","ieee":"J. Matoušek and U. Wagner, “New constructions of weak ε-nets,” <i>Discrete &#38; Computational Geometry</i>, vol. 32, no. 2. Springer, pp. 195–206, 2004.","ista":"Matoušek J, Wagner U. 2004. New constructions of weak ε-nets. Discrete &#38; Computational Geometry. 32(2), 195–206.","chicago":"Matoušek, Jiří, and Uli Wagner. “New Constructions of Weak ε-Nets.” <i>Discrete &#38; Computational Geometry</i>. Springer, 2004. <a href=\"https://doi.org/10.1007/s00454-004-1116-4\">https://doi.org/10.1007/s00454-004-1116-4</a>.","apa":"Matoušek, J., &#38; Wagner, U. (2004). New constructions of weak ε-nets. <i>Discrete &#38; Computational Geometry</i>. Springer. <a href=\"https://doi.org/10.1007/s00454-004-1116-4\">https://doi.org/10.1007/s00454-004-1116-4</a>"}}