Estimation of support of a probability density and estimation of support functionals

The problem of estimating the unknown support G [ belong ] [ R^N ] of a uniform density is considered under the assumption that the support G belongs to the class of "boundary fragments" with smooth upper surface. The minimax lower bounds for the accuracy of arbitrary estimators of G are obtained if the distance between sets is Hausdorff metric or measure of symmetric difference. The estimators of support are proposed which are optimal in the sense that they attain the convergence rate of the minimax lower bound. Similar results are proved for the problem of estima.tion of functionals of the density support.