  We investigate the Plateau and isoperimetric problems associated to
Fefferman's measure for strongly pseudoconvex real hypersurfaces in $\mathbb
C^n$ (focusing on the case $n=2$), showing in particular that the isoperimetric
problem shares features of both the euclidean isoperimetric problem and the
corresponding problem in Blaschke's equiaffine geometry in which the key
inequalities are reversed.
  The problems are invariant under constant-Jacobian biholomorphism, but we
also introduce a non-trivial modified isoperimetric quantity invariant under
general biholomorphism.
