Stochastic detection of some topological and geometric feature

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 $S \subset {\mathbb R}^d$. 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: 1. Is $S$ full dimensional? 2. If $S$ is full dimensional, is it "close to a lower dimensional set" $\mathcal{M}$? 3. If $S$ is "close to a lower dimensional $\mathcal{M}$", can we \indent a) estimate $\mathcal{M}$? \indent b) estimate some functionals defined on $\mathcal{M}$ (in particular, the Minkowski content of $\mathcal{M}$)? The theoretical results are complemented with some simulations and graphical illustrations.