## Abstract

We show that the maximum number of pairwise nonoverlapping k-rich lenses (lenses formed by at least k circles) in an arrangement of n circles in the plane is O(n^{3}/^{2} log(n/k^{3})/k^{5}/^{2} + n/k), and the sum of the degrees of the lenses of such a family (where the degree of a lens is the number of circles that form it) is O(n^{3}/^{2} log(n/k^{3})/k^{3}/^{2} + n). Two independent proofs of these bounds are given, each interesting in its own right (so we believe). The second proof gives a bound that is weaker by a polylogarithmic factor. We then show that these bounds lead to the known bound of Agarwal et al. [J. ACM, 51 (2004), pp. 139-186] and Marcus and Tardos [J. Combin. Theory Ser. A, 113 (2006), pp. 675-691] on the number of point-circle incidences in the plane. Extensions to families of more general algebraic curves and some other related problems are also considered.

Original language | English |
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Pages (from-to) | 958-974 |

Number of pages | 17 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 36 |

Issue number | 2 |

DOIs | |

State | Published - 2022 |

## Keywords

- incidence geometry
- lens cutting
- polynomial partitioning