Minimal scalings and structural properties of scalable frames

For a unit-norm frame $F = \{f_i\}_{i=1}^k$ in $\R^n$, a scaling is a vector $c=(c(1),\dots,c(k))\in \R_{\geq 0}^k$ such that $\{\sqrt{c(i)}f_i\}_{i =1}^k$ is a Parseval frame in $\R^n$. If such a scaling exists, $F$ is said to be scalable. A scaling $c$ is a minimal scaling if $\{f_i : c(i)>0\}$ has no proper scalable subframe. It is known that the set of all scalings of $F$ is a convex polytope whose vertices correspond to minimal scalings. In this paper, we provide an estimation of the number of minimal scalings of a scalable frame and a characterization of when minimal scalings are affinely dependent. Using this characterization, we can conclude that all strict scalings $c=(c(1),\dots,c(k))\in \R_{> 0}^k$ of $F$ have the same structural property. We also present the uniqueness of orthogonal partitioning property of any set of minimal scalings, which provides all possible tight subframes of a given scaled frame.