{"publication_status":"published","author":[{"first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Valtr, Pavel","last_name":"Valtr","first_name":"Pavel"},{"full_name":"Welzl, Emo","last_name":"Welzl","first_name":"Emo"}],"publist_id":"2103","date_created":"2018-12-11T12:06:29Z","_id":"4022","extern":"1","acknowledgement":"Partially supported by the National Science Foundation, under Grant ASC-9200301 and the Alan T. Waterman award, Grant CCR-9118874.","issue":"3","language":[{"iso":"eng"}],"year":"1997","type":"journal_article","page":"243 - 255","title":"Cutting dense point sets in half","citation":{"ieee":"H. Edelsbrunner, P. Valtr, and E. Welzl, “Cutting dense point sets in half,” <i>Discrete &#38; Computational Geometry</i>, vol. 17, no. 3. Springer, pp. 243–255, 1997.","apa":"Edelsbrunner, H., Valtr, P., &#38; Welzl, E. (1997). Cutting dense point sets in half. <i>Discrete &#38; Computational Geometry</i>. Springer. <a href=\"https://doi.org/10.1007/PL00009291\">https://doi.org/10.1007/PL00009291</a>","ama":"Edelsbrunner H, Valtr P, Welzl E. Cutting dense point sets in half. <i>Discrete &#38; Computational Geometry</i>. 1997;17(3):243-255. doi:<a href=\"https://doi.org/10.1007/PL00009291\">10.1007/PL00009291</a>","short":"H. Edelsbrunner, P. Valtr, E. Welzl, Discrete &#38; Computational Geometry 17 (1997) 243–255.","ista":"Edelsbrunner H, Valtr P, Welzl E. 1997. Cutting dense point sets in half. Discrete &#38; Computational Geometry. 17(3), 243–255.","mla":"Edelsbrunner, Herbert, et al. “Cutting Dense Point Sets in Half.” <i>Discrete &#38; Computational Geometry</i>, vol. 17, no. 3, Springer, 1997, pp. 243–55, doi:<a href=\"https://doi.org/10.1007/PL00009291\">10.1007/PL00009291</a>.","chicago":"Edelsbrunner, Herbert, Pavel Valtr, and Emo Welzl. “Cutting Dense Point Sets in Half.” <i>Discrete &#38; Computational Geometry</i>. Springer, 1997. <a href=\"https://doi.org/10.1007/PL00009291\">https://doi.org/10.1007/PL00009291</a>."},"status":"public","date_published":"1997-04-01T00:00:00Z","quality_controlled":"1","publication":"Discrete & Computational Geometry","publication_identifier":{"issn":["0179-5376"]},"doi":"10.1007/PL00009291","month":"04","article_processing_charge":"No","scopus_import":"1","date_updated":"2022-08-18T14:08:38Z","intvolume":"        17","article_type":"original","user_id":"ea97e931-d5af-11eb-85d4-e6957dddbf17","day":"01","volume":17,"oa_version":"None","publisher":"Springer","abstract":[{"lang":"eng","text":"A halving hyperplane of a set S of n points in R(d) contains d affinely independent points of S so that equally many of the points off the hyperplane lie in each of the two half-spaces. We prove bounds on the number of halving hyperplanes under the condition that the ratio of largest over smallest distance between any two points is at most delta n(1/d), delta some constant. Such a set S is called dense. In d = 2 dimensions the number of halving lines for a dense set can be as much as Omega(n log n), and it cannot exceed O (n(5/4)/log* n). The upper bound improves over the current best bound of O (n(3/2)/log* n) which holds more generally without any density assumption. In d = 3 dimensions we show that O (n(7/3)) is an upper bound on the number of halving planes for a dense set, The proof is based on a metric argument that can be extended to d greater than or equal to 4 dimensions, where it leads to O (n(d-2/d)) as an upper bound for the number of halving hyperplanes."}]}