  In this paper we consider two problems in frame theory. On the one hand,
given a set of vectors $\mathcal F$ we describe the spectral and geometrical
structure of optimal completions of $\mathcal F$ by a finite family of vectors
with prescribed norms, where optimality is measured with respect to
majorization. In particular, these optimal completions are the minimizers of a
family of convex functionals that include the mean square error and the
Bendetto-Fickus' frame potential. On the other hand, given a fixed frame
$\mathcal F$ we describe explicitly the spectral and geometrical structure of
optimal frames $\mathcal G$ that are in duality with $\mathcal F$ and such that
the Frobenius norms of their analysis operators is bounded from below by a
fixed constant. In this case, optimality is measured with respect to
submajorization of the frames operators. Our approach relies on the description
of the spectral and geometrical structure of matrices that minimize
submajorization on sets that are naturally associated with the problems above.
