This work is closely related to the theories of set estimation and manifold estimation. Our object of interest is a, possibly lower-dimensional, compact set S ⊂ ℝd. The general aim is to identify (via stochastic procedures) some qualitative or quantitative features of S, of geometric or topological character. The available information is just a random sample of points drawn on S. The term “to identify” means here to achieve a correct answer almost surely (a.s.) when the sample size tends to infinity. More specifically the paper aims at giving some partial answers to the following questions: is S full dimensional? Is S “close to a lower dimensional set” M? If so, can we estimate M or some functionals of M (in particular, the Minkowski content of M)? As an important auxiliary tool in the answers of these questions, a denoising procedure is proposed in order to partially remove the noise in the original data. The theoretical results are complemented with some simulations and graphical illustrations. © 2017, Institute of Mathematical Statistics.