Minimal sets in Riemannian manifolds

We study the existence of sets that minimize their measure amongst a family stable under homotopy in a compact, boundaryless Riemannian manifold. We consider the problem in the category of sets without resorting to the weakened notions of Federer (currents) or Almgren (varifolds). This setting allows to free oneself of additional regularity assumptions such as orientability, or even rectifiability of the competitors. In this talk, I will try to explain how a polyhedral approximation process inspired by Federer can be generalized to this setup and make up for the lack of compactness of the set-based approach.