  Monotone sets have been introduced about ten years ago by Cheeger and Kleiner
who reduced the proof of the non biLipschitz embeddability of the Heisenberg
group into $L^1$ to the classification of its monotone subsets. Later on,
monotone sets played an important role in several works related to geometric
measure theory issues in the Heisenberg setting. In this paper, we work in an
arbitrary Carnot group and show that its monotone subsets are sets with locally
finite perimeter that are local minimizers for the perimeter. Under an
additional condition on the ambient Carnot group, we prove that their
measure-theoretic interior and support are precisely monotone. We also prove
topological and measure-theoretic properties of local minimizers for the
perimeter whose interest is independent from the study of monotone sets. As a
combination of our results, we get in particular a sufficient condition under
which any monotone set admits measure-theoretic representatives that are
precisely monotone.
