In this paper we study some aspects of oblique duality between finite sequences of vectors $cF$ and $cG$ lying in finite dimensional subspaces $cW$ and $cV$, respectively. We compute the possible eigenvalue lists of the frame operators of oblique dua
ls to $cF$ lying in $cV$; we then compute the spectral and geometrical structure of minimizers of convex potentials among oblique duals for $cF$ under some restrictions. We obtain a complete quantitative analysis of the impact that the relative geometry between the subspaces $cV$ and $cW$ has in oblique duality. We apply this analysis to compute those rigid rotations $U$ for $cW$ such that the canonical oblique dual of $Ucdot cF$ minimize every convex potential; we also introduce a notion of aliasing for oblique dual pairs and compute those rigid rotations $U$ for $cW$ such that the canonical oblique dual pair associated to $Ucdot cF$ minimize the aliasing. We point out that these two last problems are intrinsic to the theory of oblique duality.
Given a finite sequence of vectors $mathcal F_0$ in $C^d$ we characterize in a complete and explicit way the optimal completions of $mathcal F_0$ obtained by adding a finite sequence of vectors with prescribed norms, where optimality is measured with
respect to majorization (of the eigenvalues of the frame operators of the completed sequence). Indeed, we construct (in terms of a fast algorithm) a vector - that depends on the eigenvalues of the frame operator of the initial sequence $cF_0$ and the sequence of prescribed norms - that is a minimum for majorization among all eigenvalues of frame operators of completions with prescribed norms. Then, using the eigenspaces of the frame operator of the initial sequence $cF_0$ we describe the frame operators of all optimal completions for majorization. Hence, the concrete optimal completions with prescribed norms can be obtained using recent algorithmic constructions related with the Schur-Horn theorem. The well known relation between majorization and tracial inequalities with respect to convex functions allow to describe our results in the following equivalent way: given a finite sequence of vectors $mathcal F_0$ in $C^d$ we show that the completions with prescribed norms that minimize the convex potential induced by a strictly convex function are structural minimizers, in the sense that they do not depend on the particular choice of the convex potential.
