An overview on the Cheeger problem

The Cheeger problem consists in minimizing the ratio "perimeter over volume" among subsets of a given, bounded domain. This problem has connections with several variational problems (eigenvalue estimates, prescribed mean curvature, Total Variation minimization, and others). After introducing the problem, we will focus on properties of Cheeger sets (i.e., solutions of the above minimization problem) with special emphasis on the two dimensional case. In particular, we shall present some recent results obtained in collaboration with A. Pratelli, on the characterization of Cheeger sets in non-convex, planar domains.