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 journalArticlepeer-review

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 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)].

Original languageEnglish
Pages (from-to)930-953
Number of pages24
JournalDiscrete and Computational Geometry
Volume54
Issue number4
DOIs
StatePublished - 13 Oct 2015

Funding

FundersFunder number
Hermann Minkowski-MINERVA Center for Geometry
Israel Science Foundation
Israeli Centers for Research Excellence
National Science Foundation
Natural Sciences and Engineering Research Council of Canada
Országos Tudományos Kutatási AlapprogramokNK 104183
Tel Aviv University
United States-Israel Binational Science Foundation
National Science Foundation892/13
Natural Sciences and Engineering Research Council of CanadaOTKA NK 104183, ERC-AdG 321104
United States-Israel Binational Science Foundation
Israel Science Foundation2012/229
Tel Aviv University
Israeli Centers for Research Excellence4/11

    Keywords

    • Combinatorial geometry
    • Incidences
    • Unit circles

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