On the estimation of convex polytopes

Set estimation started to interest scientists since the 1960's, even earlier, when stochastic geometry was developing much, dealing with many questions related to optimization, approximation, imaging, economics, ... Although it was studied from a probabilistic and geometric prospective, it really arose as a statistical question in the early 1990's, when boundary fragments and convex sets were investigated in the minimax framework. I will give a brief historical summary of set estimation, and I will focus on the estimation of convex polytopes. In particular I will propose new estimators that are nearly optimal in the minimax setup, and I will talk about adaptation with respect to the number of vertices of the unknown polytope.