A free lunch: manifolds of positive reach can be smoothed without decreasing the reach
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912147139723264 |
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| author | Kouřimská, Hana Dal Poz Lieutier, André Wintraecken, Mathijs |
| author_facet | Kouřimská, Hana Dal Poz Lieutier, André Wintraecken, Mathijs |
| contents | 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! |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04899 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A free lunch: manifolds of positive reach can be smoothed without decreasing the reach Kouřimská, Hana Dal Poz Lieutier, André Wintraecken, Mathijs Computational Geometry 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! |
| title | A free lunch: manifolds of positive reach can be smoothed without decreasing the reach |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2412.04899 |