Universally consistent estimation of the reach

The reach of a set $M \subset \mathbb R^d$, also known as condition number when $M$ is a manifold, was introduced by Federer in 1959 and is a central concept in geometric measure theory, set estimation, manifold learning, among others areas. We introduce a universally consistent estimate of the reach, just assuming that the reach is positive. A necessary condition for the universal convergence of the reach is that the Haussdorf distance between the sample and the set converges to zero. Without further assumptions we show that the convergence rate of this distance can be arbitrarily slow. However, under a weak additional assumption, we provide rates of convergence for the reach estimator. We also show that it is not possible to determine if the reach of the support of a density is zero or not based on a finite sample. We provide a small simulation study and a bias correction method for the case when $M$ is a manifold.