Methods in Estimation of Convex Sets

In the framework of shape constrained estimation, we review methods and works done in convex set estimation. Some methods are standard in statistics, coming especially from the theory of empirical processes, but we more geometric methods that are specific to convex set estimation. The statistical problems that we review include density support estimation, estimation of the level sets of densities or depth functions, nonparametric regression, etc. We focus on the estimation of convex sets under the Nykodim and the Hausdorff metrics, which require different techniques and, quite surprisingly, lead to very different results, in particular in density support estimation. Finally, we discuss computational issues in high dimension.