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

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