{"abstract":[{"text":"This paper proves that any set of n points in the plane contains two points such that any circle through those two points encloses at least n12−112+O(1)n47 points of the set. The main ingredients used in the proof of this result are edge counting formulas for k-order Voronoi diagrams and a lower bound on the minimum number of semispaces of size at most k.","lang":"eng"}],"publication":"Geometriae Dedicata","page":"1 - 12","publist_id":"2043","publication_status":"published","date_published":"1989-10-01T00:00:00Z","date_updated":"2019-04-26T07:22:40Z","date_created":"2018-12-11T12:06:49Z","issue":"1","doi":"10.1007/BF00181432","day":"01","_id":"4080","author":[{"last_name":"Edelsbrunner","full_name":"Herbert Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","first_name":"Herbert"},{"first_name":"Nany","full_name":"Hasan, Nany","last_name":"Hasan"},{"first_name":"Raimund","full_name":"Seidel, Raimund","last_name":"Seidel"},{"first_name":"Xiao","full_name":"Shen, Xiao-Jun","last_name":"Shen"}],"title":"Circles through two points that always enclose many points","month":"10","year":"1989","quality_controlled":0,"publisher":"Kluwer","type":"journal_article","citation":{"chicago":"Edelsbrunner, Herbert, Nany Hasan, Raimund Seidel, and Xiao Shen. “Circles through Two Points That Always Enclose Many Points.” Geometriae Dedicata 32, no. 1 (1989): 1–12. https://doi.org/10.1007/BF00181432.","ama":"Edelsbrunner H, Hasan N, Seidel R, Shen X. Circles through two points that always enclose many points. Geometriae Dedicata. 1989;32(1):1-12. doi:10.1007/BF00181432","mla":"Edelsbrunner, Herbert, et al. “Circles through Two Points That Always Enclose Many Points.” Geometriae Dedicata, vol. 32, no. 1, Kluwer, 1989, pp. 1–12, doi:10.1007/BF00181432.","short":"H. Edelsbrunner, N. Hasan, R. Seidel, X. Shen, Geometriae Dedicata 32 (1989) 1–12.","apa":"Edelsbrunner, H., Hasan, N., Seidel, R., & Shen, X. (1989). Circles through two points that always enclose many points. Geometriae Dedicata, 32(1), 1–12. https://doi.org/10.1007/BF00181432","ista":"Edelsbrunner H, Hasan N, Seidel R, Shen X. 1989. Circles through two points that always enclose many points. Geometriae Dedicata. 32(1), 1–12.","ieee":"H. Edelsbrunner, N. Hasan, R. Seidel, and X. Shen, “Circles through two points that always enclose many points,” Geometriae Dedicata, vol. 32, no. 1, pp. 1–12, 1989."},"extern":1,"volume":32,"intvolume":" 32","acknowledgement":"Work on this paper by the first author has been supported by Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862 and by the National Science Foundation under Grant CCR-8714565, by the second author has been partially supported by the Digital Equipment Corporation, by the fourth author has been partially supported by the Office of Naval Research under Grant N00014-86K-0416.","status":"public"}