A new method for smoothing a multivariate data set is introduced that is based on a simple geometric operation. This method is applied to the problem of estimating level sets of a density and minimum volume sets with given probability content. Building on existing techniques, the resulting estimator combines excellent theoretical and computational properties: It converges with the minimax rates (up to log factors) in most cases where these rates are known and, at the same time, it can be computed, visualized, stored and manipulated by simple algorithms and tools. It is applicable to a wide class of sets that is characterized explicitly in terms of the underlying densities and includes nonconvex and disconnected sets, and it is argued that it should give reasonable results in completely general situations. Applications to the construction of multivariate confidence regions in frequentist and Bayesian contexts are briefly mentioned.