{"title":"Cutting dense point sets in half","article_processing_charge":"No","publist_id":"2103","author":[{"orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert"},{"full_name":"Valtr, Pavel","last_name":"Valtr","first_name":"Pavel"},{"first_name":"Emo","last_name":"Welzl","full_name":"Welzl, Emo"}],"user_id":"ea97e931-d5af-11eb-85d4-e6957dddbf17","citation":{"short":"H. Edelsbrunner, P. Valtr, E. Welzl, Discrete & Computational Geometry 17 (1997) 243–255.","ieee":"H. Edelsbrunner, P. Valtr, and E. Welzl, “Cutting dense point sets in half,” Discrete & Computational Geometry, vol. 17, no. 3. Springer, pp. 243–255, 1997.","ama":"Edelsbrunner H, Valtr P, Welzl E. Cutting dense point sets in half. Discrete & Computational Geometry. 1997;17(3):243-255. doi:10.1007/PL00009291","apa":"Edelsbrunner, H., Valtr, P., & Welzl, E. (1997). Cutting dense point sets in half. Discrete & Computational Geometry. Springer. https://doi.org/10.1007/PL00009291","mla":"Edelsbrunner, Herbert, et al. “Cutting Dense Point Sets in Half.” Discrete & Computational Geometry, vol. 17, no. 3, Springer, 1997, pp. 243–55, doi:10.1007/PL00009291.","ista":"Edelsbrunner H, Valtr P, Welzl E. 1997. Cutting dense point sets in half. Discrete & Computational Geometry. 17(3), 243–255.","chicago":"Edelsbrunner, Herbert, Pavel Valtr, and Emo Welzl. “Cutting Dense Point Sets in Half.” Discrete & Computational Geometry. Springer, 1997. https://doi.org/10.1007/PL00009291."},"date_created":"2018-12-11T12:06:29Z","date_published":"1997-04-01T00:00:00Z","doi":"10.1007/PL00009291","page":"243 - 255","publication":"Discrete & Computational Geometry","day":"01","year":"1997","quality_controlled":"1","publisher":"Springer","acknowledgement":"Partially supported by the National Science Foundation, under Grant ASC-9200301 and the Alan T. Waterman award, Grant CCR-9118874.","extern":"1","date_updated":"2022-08-18T14:08:38Z","status":"public","type":"journal_article","article_type":"original","_id":"4022","issue":"3","volume":17,"language":[{"iso":"eng"}],"publication_status":"published","publication_identifier":{"issn":["0179-5376"]},"intvolume":" 17","month":"04","scopus_import":"1","oa_version":"None","abstract":[{"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.","lang":"eng"}]}