  The recently introduced and characterized scalable frames can be considered
as those frames which allow for perfect preconditioning in the sense that the
frame vectors can be rescaled to yield a tight frame. In this paper we define
$m$-scalability, a refinement of scalability based on the number of non-zero
weights used in the rescaling process, and study the connection between this
notion and elements from convex geometry. Finally, we provide results on the
topology of scalable frames. In particular, we prove that the set of scalable
frames with "small" redundancy is nowhere dense in the set of frames.
