Computable bounds for the reach and $r$-convexity of subsets of $\mathbb{R}^d$

TL;DR

This paper develops methods to compute upper bounds on the reach and r-convexity of compact sets in Euclidean space from point cloud data, with convergence guarantees and applications to high-dimensional geometry.

Contribution

It introduces computable bounds for reach and r-convexity from point clouds, including the novel $eta$-reach, with convergence analysis and practical implications.

Findings

Bounds converge to true geometric quantities as data density increases.

The rate of convergence for the reach bound is established under weak regularity.

Numerical studies demonstrate the use of $eta$-reach in high-dimensional manifold inference.

Abstract

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. 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 point cloud becomes dense in the set, 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.