On Triple Intersections of Three Families of Unit Circles

Orit E. Raz; Micha Sharir; József Solymosi

doi:10.1007/s00454-015-9734-6

On Triple Intersections of Three Families of Unit Circles

Orit E. Raz^*, Micha Sharir, József Solymosi

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

Let p₁, p₂, p₃ be three distinct points in the plane, and, for i = 1, 2, 3, let C_i be a family of n unit circles that pass through p_i. We address a conjecture made by Székely, and show that the number of points incident to a circle of each family is O(n^11/6), improving an earlier bound for this problem due to Elekes et al. (Comb Probab Comput 18:691–705, 2009). The problem is a special instance of a more general problem studied by Elekes and Szabó (Combinatorica 32:537–571, 2012) [and by Elekes and Rónyai (J Comb Theory Ser A 89:1–20, 2000)].

Original language	American English
Pages (from-to)	930-953
Number of pages	24
Journal	Discrete and Computational Geometry
Volume	54
Issue number	4
DOIs	https://doi.org/10.1007/s00454-015-9734-6
State	Published - 13 Oct 2015
Externally published	Yes

Bibliographical note

Funding Information:
We would like to thank an anonymous referee, whose helpful and constructive comments helped us a lot to improve the paper. Part of this research was performed while the authors were visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation. The authors deeply appreciate the stimulating environment and facilities provided by IPAM. Work on this paper by Orit E. Raz and Micha Sharir was supported by Grant 892/13 from the Israel Science Foundation. Work by Micha Sharir was also supported by Grant 2012/229 from the U.S.–Israel Binational Science Foundation, by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. Work by József Solymosi was supported by NSERC, ERC-AdG 321104, and OTKA NK 104183 grants.

Publisher Copyright:
© 2015, Springer Science+Business Media New York.

Keywords

Combinatorial geometry
Incidences
Unit circles

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10.1007/s00454-015-9734-6

Cite this

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title = "On Triple Intersections of Three Families of Unit Circles",

abstract = "Let p1, p2, p3 be three distinct points in the plane, and, for i = 1, 2, 3, let Ci be a family of n unit circles that pass through pi. We address a conjecture made by Sz{\'e}kely, and show that the number of points incident to a circle of each family is O(n11/6), improving an earlier bound for this problem due to Elekes et al. (Comb Probab Comput 18:691–705, 2009). The problem is a special instance of a more general problem studied by Elekes and Szab{\'o} (Combinatorica 32:537–571, 2012) [and by Elekes and R{\'o}nyai (J Comb Theory Ser A 89:1–20, 2000)].",

keywords = "Combinatorial geometry, Incidences, Unit circles",

author = "Raz, {Orit E.} and Micha Sharir and J{\'o}zsef Solymosi",

note = "Funding Information: We would like to thank an anonymous referee, whose helpful and constructive comments helped us a lot to improve the paper. Part of this research was performed while the authors were visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation. The authors deeply appreciate the stimulating environment and facilities provided by IPAM. Work on this paper by Orit E. Raz and Micha Sharir was supported by Grant 892/13 from the Israel Science Foundation. Work by Micha Sharir was also supported by Grant 2012/229 from the U.S.–Israel Binational Science Foundation, by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. Work by J{\'o}zsef Solymosi was supported by NSERC, ERC-AdG 321104, and OTKA NK 104183 grants. Publisher Copyright: {\textcopyright} 2015, Springer Science+Business Media New York.",

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N2 - Let p1, p2, p3 be three distinct points in the plane, and, for i = 1, 2, 3, let Ci be a family of n unit circles that pass through pi. We address a conjecture made by Székely, and show that the number of points incident to a circle of each family is O(n11/6), improving an earlier bound for this problem due to Elekes et al. (Comb Probab Comput 18:691–705, 2009). The problem is a special instance of a more general problem studied by Elekes and Szabó (Combinatorica 32:537–571, 2012) [and by Elekes and Rónyai (J Comb Theory Ser A 89:1–20, 2000)].

AB - Let p1, p2, p3 be three distinct points in the plane, and, for i = 1, 2, 3, let Ci be a family of n unit circles that pass through pi. We address a conjecture made by Székely, and show that the number of points incident to a circle of each family is O(n11/6), improving an earlier bound for this problem due to Elekes et al. (Comb Probab Comput 18:691–705, 2009). The problem is a special instance of a more general problem studied by Elekes and Szabó (Combinatorica 32:537–571, 2012) [and by Elekes and Rónyai (J Comb Theory Ser A 89:1–20, 2000)].

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ER -

On Triple Intersections of Three Families of Unit Circles

Abstract

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Keywords

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