  The selection of basic variables in current-density functional theory and
formal properties of the resulting formulations are critically examined. Focus
is placed on the extent to which the Hohenberg--Kohn theorem,
constrained-search approach and Lieb's formulation (in terms of convex and
concave conjugation) of standard density-functional theory can be generalized
to provide foundations for current-density functional theory. For the
well-known case with the gauge-dependent paramagnetic current density as a
basic variable, we find that the resulting total energy functional is not
concave. It is shown that a simple redefinition of the scalar potential
restores concavity and enables the application of convex analysis and
convex/concave conjugation. As a result, the solution sets arising in
potential-optimization problems can be given a simple characterization. We also
review attempts to establish theories with the physical current density as a
basic variable. Despite the appealing physical motivation behind this choice of
basic variables, we find that the mathematical foundations of the theories
proposed to date are unsatisfactory. Moreover, the analogy to standard
density-functional theory is substantially weaker as neither the
constrained-search approach nor the convex analysis framework carry over to a
theory making use of the physical current density.
