  We investigate relations between spectral properties of a single-centre
point-interaction Hamiltonian describing a particle confined to a bounded
domain $\Omega\subset\mathbb{R}^{d},\: d=2,3$, with Dirichlet boundary, and the
geometry of $\Omega$. For this class of operators Krein's formula yields an
explicit representation of the resolvent in terms of the integral kernel of the
unperturbed one, $(-\Delta_{\Omega}^{D}+z) ^{-1}$. We use a moving plane
analysis to characterize the behaviour of the ground-state energy of the
Hamiltonian with respect to the point-interaction position and the shape of
$\Omega$, in particular, we establish some conditions showing how to place the
interaction to optimize the principal eigenvalue.
