  Federer's characterization states that a set $E\subset \mathbb{R}^n$ is of
finite perimeter if and only if $\mathcal H^{n-1}(\partial^*E)<\infty$. Here
the measure-theoretic boundary $\partial^*E$ consists of those points where
both $E$ and its complement have positive upper density. We show that the
characterization remains true if $\partial^*E$ is replaced by a smaller
boundary consisting of those points where the \emph{lower} densities of both
$E$ and its complement are at least a given number. This result is new even in
Euclidean spaces but we prove it in a more general complete metric space that
is equipped with a doubling measure and supports a Poincar\'e inequality.
