Let $(M,g)$ be a Riemannian manifold and consider a Poisson process generating, in average, $n$ uniformly distributed points in $(M,g)$. In joint work with Omer Bobrowski we answer the following question: As the number, $n$, of points increases what is the smallest possible radius $r$, so that the union of the radius $r$ Riemannian balls centered at the randomly generated points has the same homology as that of the underlying Riemannian manifold $M$.

In terms of a data set lying in a Riemannian manifold, this is similar to asking: what is the minimum we must fatten the data points so that they recover the underlying topology being encoded.