We consider the problem of estimating a set S from a random sample of points of S, which amounts at estimating the support of the underlying probability density. Set estimation has applications in various situations, including medical diagnosises, image analysis, and quality control for example. We focus on the simple set estimator defined as the union of balls centered at the random points. Using tools from Riemannian geometry, and under mild analytic conditions on the underlying density of the data, we derive the exact rate of convergence of this set estimator. In closed connection with the problem of set estimation, we study the estimation of the number of connected components of a level set of a multivariate probability density. This allows one to assess the number of clusters of a statistical population, which is an essential problem of unsupervised learning. We introduce an estimator based on a graph, and using similar geometrical tools, we establish the asymptotic consistency of the methodology.