A free lunch: manifolds of positive reach can be smoothed without decreasing the reach

TL;DR

This paper proves that manifolds with positive reach can be smoothly approximated without reducing their reach, enabling the extension of many geometric algorithms' guarantees to less smooth manifolds.

Contribution

It introduces a method to approximate manifolds with positive reach by smooth manifolds without decreasing reach, broadening applicability of existing geometric results.

Findings

Any manifold with positive reach can be approximated by a smooth manifold with similar reach.

The approximation preserves reach while improving smoothness to C^.

Many theorems for C^2 manifolds extend to less smooth manifolds without loss of reach.

Abstract

Assumptions on the reach are crucial for ensuring the correctness of many geometric and topological algorithms, including triangulation, manifold reconstruction and learning, homotopy reconstruction, and methods for estimating curvature or reach. However, these assumptions are often coupled with the requirement that the manifold be smooth, typically at least C^2 .In this paper, we prove that any manifold with positive reach can be approximated arbitrarily well by a C^$\infty$ manifold without significantly reducing the reach, by employing techniques from differential topology -partitions of unity and smoothing using convolution kernels. This result implies that nearly all theorems established for C^2 manifolds with a certain reach naturally extend to manifolds with the same reach, even if they are not C^2 , for free!