  Let $S\in\mathcal{M}_d(\mathbb{C})^+$ be a positive semidefinite $d\times d$
complex matrix and let $\mathbf a=(a_i)_{i\in\mathbb{I}_k}\in
\mathbb{R}_{>0}^k$, indexed by $\mathbb{I}_k=\{1,\ldots,k\}$, be a $k$-tuple of
positive numbers. Let $\mathbb T_{d}(\mathbf a )$ denote the set of families
$\mathcal G=\{g_i\}_{i\in\mathbb{I}_k}\in (\mathbb{C}^d)^k$ such that
$\|g_i\|^2=a_i$, for $i\in\mathbb{I}_k$; thus, $\mathbb T_{d}(\mathbf a )$ is
the product of spheres in $\mathbb{C}^d$ endowed with the product metric. For a
strictly convex unitarily invariant norm $N$ in $\mathcal{M}_d(\mathbb{C})$, we
consider the generalized frame operator distance function $\Theta_{( N \, , \,
S\, , \, \mathbf a)}$ defined on $\mathbb T_{d}(\mathbf a )$, given by $$
\Theta_{( N \, , \, S\, , \, \mathbf a)}(\mathcal G) =N(S-S_{\mathcal G })
\quad \text{where} \quad S_{\mathcal G}=\sum_{i\in\mathbb{I}_k}
g_i\,g_i^*\in\mathcal{M}_d(\mathbb{C})^+\,. $$ In this paper we determine the
geometrical and spectral structure of local minimizers $\mathcal G_0\in\mathbb
T_{d}(\mathbf a )$
  of $\Theta_{( N \, , \, S\, , \, \mathbf a)}$. In particular, we show that
local minimizers are global minimizers, and that these families do not depend
on the particular choice of $N$.
