  Given a convex domain $C$, a $C$-polygon is an intersection of $n\geq 2$
homothets of $C$. If the homothets are translates of $C$ then we call the
intersection a translative $C$-polygon. This paper proves that if $C$ is a
strictly convex domain with $m$ singular boundary points, then the number of
singular boundary points a $C$-polygon has is between $n$ and $2(n-1)+m$. For a
translative $C$-polygon we show the number of singular boundary points is
between $n$ and $n+m$.
