Isoperimetric inequality for nearly spherical domains in the Bergman ball

We prove a quantitative isoperimetric inequality for the nearly spherical subset of the Bergman ball in ${\mathbb{C}}^n$. We prove the Fuglede theorem for such sets. This result is a counterpart of a similar result obtained for the hyperbolic unit ball and it makes the first result on the isoperimetric phenomenon in the Bergman ball.