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

TL;DR

The paper proves that any compact manifold in $\mathbb{R}^d$ with positive reach $R$ can be arbitrarily closely approximated by a $C^\infty$ embedded submanifold without substantial loss of reach. By locally representing the manifold as a graph of a $C^{1,1}$ map from the tangent to normal space and then smoothing via kernel convolution within a partition-of-unity framework, the authors control Lipschitz constants and derivative bounds. Federer’s reach characterization is used to quantify how smoothing affects the reach, with careful bounds on tangent-space angles and local Hausdorff distances, enabling a global $C^1$-proximate smoothing that preserves reach up to $\varepsilon$. The result implies that dense, smooth approximations exist for the class of positive-reach manifolds, allowing the transfer of $C^2$-manifold results to broader, non-smooth settings for free, with potential impact on geometric algorithms such as triangulation, reconstruction, and curvature estimation. Overall, the work provides a constructive smoothing pathway that preserves geometric integrity while increasing regularity. $$\text{rch}(\mathcal{M}')\ge R-\varepsilon, \quad \mathcal{M}'\to\mathcal{M}\text{ in }C^1\text{, and } \mathcal{M}'\text{ is }C^\infty.$$

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!

Figures (15)

• Figure 1: A $C^{1,1}$ transition (in red) between a circular arc and a straight line segment.
• Figure 2: In our setting, we view manifolds locally as graphs of functions.
• Figure 3: First we cover the manifold $\mathcal{M}$ with neighbourhoods in each of which $\mathcal{M}$ is representable as a graph of a function.
• Figure 4: The function $F$ (red) is smooth on the set $U_1$ (green) and equals $f$ on the set $U_3$ (blue).
• Figure 5: We construct the manifold $\mathcal{M}'$ iteratively. In each neighbourhood (gray), we replace the original manifold ($(U,f(U))$, in black) by a smooth piece ($(U,F(U))$, in red).

Theorems & Definitions (25)

• Theorem 1
• Remark 2
• Corollary 3
• Theorem 4: adapted from Theorems 1 and 4 and Lemmas 33 and 37 of CompanionPaperC11
• Theorem 5: adapted from Theorem 4.18 of Federer
• Definition 6
• Theorem 7: Theorem 2.1 of Hirsch1976
• Definition 8: Smoothing kernel
• Definition 9
• Definition 10: Convolution