  We show that the relative entropy between the reduced density matrix of the
vacuum state in some region $A$ and that of an excited state created by a
unitary operator localized at a small distance $\ell$ of a boundary point $p$
is insensitive to the global shape of $A$, up to a small correction. This
correction tends to zero as $\ell/R$ tends to zero, where $R$ is a measure of
the curvature of $\partial A$ at $p$, but at a rate necessarily slower than
$\sim \sqrt{\ell/R}$ (in any dimension). Our arguments are mathematically
rigorous and only use model-independent, basic assumptions about quantum field
theory such as locality and Poincare invariance.
