Computable Bounds for the Reach and r-Convexity of Subsets of $${{\mathbb {R}}}^d$$

Abstract

The convexity of a set can be generalized to the two weaker notions of positive reach and r-convexity; both describe the regularity of a set’s boundary. For any compact subset of \({{\mathbb {R}}}^d\), we provide methods for computing upper bounds on these quantities from point cloud data. The bounds converge to the respective quantities as the sampling scale of the point cloud decreases, and the rate of convergence for the bound on the reach is given under a weak regularity condition. We also introduce the \(\beta \)-reach, a generalization of the reach that excludes small-scale features of size less than a parameter \(\beta \in [0,\infty )\). Numerical studies suggest how the \(\beta \)-reach can be used in high-dimension to infer the reach and other geometric properties of smooth submanifolds.