  We obtain a trace Hardy inequality for the Euclidean space with a bounded cut
$\Sigma\subset\mathbb R^d$, $d \ge 2$. In this novel geometric setting, the
Hardy-type inequality non-typically holds also for $d = 2$. The respective
Hardy weight is given in terms of the geodesic distance to the boundary of
$\Sigma$. We provide its applications to the heat equation on $\mathbb R^d$
with an insulating cut at $\Sigma$ and to the Schr\"odinger operator with a
$\delta'$-interaction supported on $\Sigma$. We also obtain generalizations of
this trace Hardy inequality for a class of unbounded cuts.
