  We study the minimum sets of plurisubharmonic functions with strictly
positive Monge-Amp\`ere densities. We investigate the relationship between
their Hausdorff dimension and the regularity of the function. Under suitable
assumptions we prove that the minimum set cannot contain analytic subvarieties
of large dimension. In the planar case we analyze the influence on the
regularity of the right hand side and consider the corresponding free boundary
problem with irregular data. We provide sharp examples for the Hausdorff
dimension of the minimum set and the related free boundary. We also draw
several analogues with the corresponding real results.
