  In this paper, we give upper and lower bounds on the number of Steiner points
required to construct a strictly convex quadrilateral mesh for a planar point
set. In particular, we show that $3{\lfloor\frac{n}{2}\rfloor}$ internal
Steiner points are always sufficient for a convex quadrilateral mesh of $n$
points in the plane. Furthermore, for any given $n\geq 4$, there are point sets
for which $\lceil\frac{n-3}{2}\rceil-1$ Steiner points are necessary for a
convex quadrilateral mesh.
