On Triple Intersections of Three Families of Unit Circles

Orit E. Raz, Micha Sharir, József Solymosi

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

Keywords

  • Combinatorial geometry
  • Incidences
  • Unit circles

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