A data-adaptive method for estimating density level sets under shape conditions

Given a random sample of points from some unknown density, we propose a method for estimating density level sets, for a given threshold t, under the r ́convexity assumption. This shape condition generalizes the convexity property and allows to consider level sets with more than one connected component. The main problem in practice is that r is an unknown geometric characteristic of the set related to its curvature, which may depend on t. A stochastic algorithm is proposed for selecting its value from data. The resulting reconstruction of the level set is able to achieve minimax rates for Hausdorff metric and distance in measure uniformly on the level t.