On rich lenses in planar arrangements of circles and related problems

Abstract

We show that the maximum number of pairwise non-overlapping 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 )+ 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+ n . Two independent proofs of these bounds are given, each interesting in its own right (so we believe). We then show that these bounds lead to the known bound of [1,8] 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.