A note on lenses in arrangements of pairwise intersecting circles in the plane

Abstract

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.