  The many-body problem can in general not be solved exactly, and one of the
most prominent approximations is to build perturbation expansions. A huge
variety of expansions is possible, which differ by the quantity to be expanded,
the expansion variable, the starting point, and ideas how to resum or terminate
the series. Although much has been discussed and much has been done, some
choices were made for historical reasons, in particular, limited computation or
storage capacities. The present work aims at examining the justifications for
different choices made in different contexts, by comparing ingredients of
functionals based on \GF s on one side, and on the charge density on the other
side. Of particular interest will be the question of how to build an optimal
starting point for the approximation of non-local quantities, making use of
near- or far-sightedness, and daring to consider models beyond the homogeneous
electron gas. This will include the use of connector approximations. We will
also discuss why it is a good idea to build functionals of the density.
