We study the problem of estimating a set G in IR (or, equivalently, estimating its boundary) given n independent identically distributed in G observations X^,...,X. We suppose that the boundary of G can be represented as a monotone or convex function of k-1 arguments. We evaluate the risks of several estimators of boundaries and show that they converge with the best possible rates. A general density of X.'s with the support G is considered, as well as the extension to the case where a small portion of "data outliers" falls out of the set G.