Isoperimetric inequality for nearly spherical domains in the Bergman ball double struck upper B 2$\mathbb{B}_2$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://w

<jats:p> We prove a quantitative isoperimetric inequality for nearly spherical domains in the Bergman ball in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" content-type="simple" xlink:href="S0308210526101632_inline2.png"> <jats:alt-text content-type="machine-generated">double struck upper C 2</jats:alt-text> </jats:inline-graphic> <jats:tex-math>$\mathbb{C}^2$</jats:tex-math> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:msup> <mml:mi>ℂ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . We prove a Fuglede-type theorem for such sets. This result is a counterpart of a similar result obtained for the hyperbolic unit ball, and it provides the first result on the isoperimetric phenomenon in the Bergman ball. </jats:p>