  We show that entropy is globally concave with respect to energy for a rich
class of mean field interactions, including regularizations of the the
point-vortex model in the plane, plasmas and self-gravitating matter in 2D, as
well as the higher dimensional logarithmic interactions appearing in conformal
geometry and power laws. The proofs are based on a corresponding "microscopic"
concavity result at finite N, shown by leveraging an unexpected link to Kahler
geometry and plurisubharmonic functions. Under more restrictive homogeneity
assumptions strict concavity is obtained using a uniqueness result for free
energy minimizers, established in a companion paper. The results imply that
thermodynamic equivalence of ensembles holds for this class of mean field
models. As an application it is shown that the critical inverse negative
temperatures - in the macroscopic as well as the microscopic setting - coincide
with the asymptotic slope of the corresponding microcanonical entropies. Along
the way we also extend previous results on the thermodynamic equivalence of
ensembles for continuous weakly positive definite interactions, concerning
positive temperature states, to the general non-continuous case. In particular,
singular situations are exhibited where, somewhat surprisingly, thermodynamic
equivalence of ensembles fails at energy levels sufficiently close to the
minimum energy level.
