  We introduce an extension of the convex potentials for finite frames (e.g.
the frame potential defined by Benedetto and Fickus) in the framework of Bessel
sequences of integer translates of finite sequences in $L^2(\R^k)$. We show
that under a natural normalization hypothesis, these convex potentials detect
tight frames as their minimizers. We obtain a detailed spectral analysis of the
frame operators of shift generated oblique duals of a fixed frame of
translates. We use this result to obtain the spectral and geometrical structure
of optimal shift generated oblique duals with norm restrictions, that
simultaneously minimize every convex potential; we approach this problem by
showing that the water-filling construction in probability spaces is optimal
with respect to submajorization (within an appropriate set of functions) and by
considering a non-commutative version of this construction for measurable
fields of positive operators.
