Identifying the reach from high-dimensional point cloud data with connections to r-convexity

The convexity of a set can be generalized to the two weaker notions of reach and $r$-convexity; both describe the regularity of a set's boundary. In this article, these two notions are shown to be equivalent for closed subsets of $\mathbb{R}^d$ with $C^1$ smooth, $(d-1)$-dimensional boundary. In the general case, for closed subsets of $\mathbb{R}^d$, we detail a new characterization of the reach in terms of the distance-to-set function applied to midpoints of pairs of points in the set. For compact subsets of $\mathbb{R}^d$, we provide methods of approximating the reach and $r$-convexity based on high-dimensional point cloud data. These methods are intuitive and highly tractable, and produce upper bounds that converge to the respective quantities as the density of the point cloud is increased. Simulation studies suggest that the rates at which the approximation methods converge correspond to those established theoretically.