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

Abstract

Let ${\cal{F}}$ 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 ${\cal{F}}$ is at most $2n-2$. This bound is tight. Furthermore, if no two circles in ${\cal{F}}$ touch, then the geometric graph $G$ on the set of centers of the circles in ${\cal{F}}$ whose edges correspond to the lenses generated by ${\cal{F}}$ 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.