  We show that the Poincar\'{e} inequality holds on an open set
$D\subset\mathbb{R}^n$ if and only if $D$ admits a smooth, bounded function
whose Laplacian has a positive lower bound on $D$. Moreover, we prove that the
existence of such a bounded, strictly subharmonic function on $D$ is equivalent
to the finiteness of the strict inradius of $D$ measured with respect to the
Newtonian capacity. We also obtain a sharp upper bound, in terms of this notion
of inradius, for the smallest eigenvalue of the Dirichlet--Laplacian.
