COUNTING and CUTTING RICH LENSES in ARRANGEMENTS of CIRCLES

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(n³/² log(n/k³)/k⁵/² + 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(n³/² log(n/k³)/k³/² + 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.