  In the framework of shape constrained estimation, we review methods and works
done in convex set estimation. These methods mostly build on stochastic and
convex geometry, empirical process theory, functional analysis, linear
programming, extreme value theory, etc. 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 Nikodym and 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 dimensions.
