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!

翻译：触及性假设对于确保许多几何与拓扑算法的正确性至关重要，包括三角剖分、流形重建与学习、同伦重构，以及估计曲率或触及的方法。然而，这些假设通常与流形需具备光滑性（通常至少为C^2类）的要求相结合。本文通过运用微分拓扑中的单位分解与卷积核光滑化技术，证明任何具有正触及的流形均可被C^∞流形任意逼近，且不会显著降低触及。这一结果表明，几乎所有针对具有特定触及的C^2流形建立的定理，均可自然地推广至具有相同触及的非C^2流形，且无需额外代价！