  For some special window functions $\psi_{\beta} \in H^2(\mathbb{C}^+),$ we
prove that, over all sets $\Delta \subset \mathbb{C}^+$ of fixed hyperbolic
measure $\nu(\Delta),$ the ones over which the Wavelet transform
$W_{\overline{\psi_{\beta}}}$ with window $\overline{\psi_{\beta}}$
concentrates optimally are exactly the discs with respect to the
pseudohyperbolic metric of the upper half space. This answers a question raised
by Abreu and D\"orfler.
  Our techniques make use of a framework recently developed in a previous work
by F. Nicola and the second author, but in the hyperbolic context induced by
the dilation symmetry of the Wavelet transform. This leads us naturally to use
a hyperbolic rearrangement function, as well as the hyperbolic isoperimetric
inequality, in our analysis.
