{"extern":1,"publisher":"ACM","abstract":[{"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. [7], that every point set X in Rd admits a weak ε-net of cardinality O(ε-d polylog(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)). We also prove, by a different method, a near-linear upper bound for points uniformly distributed on the (d - 1)-dimensional sphere.","lang":"eng"}],"publist_id":"4502","conference":{"name":"SoCG: Symposium on Computational Geometry"},"citation":{"chicago":"Matoušek, Jiří, and Uli Wagner. “New Constructions of Weak Epsilon-Nets,” 129–35. ACM, 2003. <a href=\"https://doi.org/10.1145/777792.777813\">https://doi.org/10.1145/777792.777813</a>.","ieee":"J. Matoušek and U. Wagner, “New constructions of weak epsilon-nets,” presented at the SoCG: Symposium on Computational Geometry, 2003, pp. 129–135.","ama":"Matoušek J, Wagner U. New constructions of weak epsilon-nets. In: ACM; 2003:129-135. doi:<a href=\"https://doi.org/10.1145/777792.777813\">10.1145/777792.777813</a>","ista":"Matoušek J, Wagner U. 2003. New constructions of weak epsilon-nets. SoCG: Symposium on Computational Geometry, 129–135.","mla":"Matoušek, Jiří, and Uli Wagner. <i>New Constructions of Weak Epsilon-Nets</i>. ACM, 2003, pp. 129–35, doi:<a href=\"https://doi.org/10.1145/777792.777813\">10.1145/777792.777813</a>.","short":"J. Matoušek, U. Wagner, in:, ACM, 2003, pp. 129–135.","apa":"Matoušek, J., &#38; Wagner, U. (2003). New constructions of weak epsilon-nets (pp. 129–135). Presented at the SoCG: Symposium on Computational Geometry, ACM. <a href=\"https://doi.org/10.1145/777792.777813\">https://doi.org/10.1145/777792.777813</a>"},"publication_status":"published","doi":"10.1145/777792.777813","author":[{"first_name":"Jiří","full_name":"Matoušek, Jiří","last_name":"Matoušek"},{"full_name":"Uli Wagner","last_name":"Wagner","first_name":"Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-1494-0568"}],"date_published":"2003-06-01T00:00:00Z","year":"2003","day":"01","type":"conference","status":"public","page":"129 - 135","month":"06","date_updated":"2021-01-12T06:57:24Z","date_created":"2018-12-11T11:57:34Z","title":"New constructions of weak epsilon-nets","_id":"2423","quality_controlled":0}