A Note on Lenses in Arrangements of Pairwise Intersecting Circles in the Plane

Let F be a family of n pairwise intersecting circles in the plane. We show that the number of lenses, that is convex digons, in the arrangement induced by F is at most 2n − 2. This bound is tight. Furthermore, if no two circles in F touch, then the geometric graph G on the set of centers of the circles in F whose edges correspond to the lenses generated by F does not contain pairs of avoiding edges. That is, G does not contain pairs of edges that are opposite edges in a convex quadrilateral. Such graphs are known to have at most 2n − 2 edges.