In this paper we consider two problems of frame theory. On the one hand,
given a fixed frame ${\mathcal F}$ we describe explicitly the spectral and
geometric structure of optimal frames ${\mathcal W}$ 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, where optimality is measured with
respect to submajorization.