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A Free Lunch: Manifolds of Positive Reach Can Be Smoothed Without Decreasing the Reach

Authors Hana Dal Poz Kouřimská , André Lieutier , Mathijs Wintraecken

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LIPIcs.SoCG.2026.37.pdf
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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Reach
  • Manifolds
  • Smoothing
  • Differentiability
  • Differential topology

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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 C².
In this paper, we prove that any manifold with positive reach can be approximated arbitrarily well by a C^∞ manifold without significantly reducing the reach. More precisely, given a manifold with reach R, we construct a manifold that is ε-close to it in the C¹ sense (both the manifold and its tangent spaces are close), and has reach at least R-ε. The proof employs techniques from differential topology - partitions of unity and smoothing using convolution kernels. 
This result implies that nearly all theorems established for C² or manifolds with a certain reach naturally extend to manifolds with the same reach, even if they are not C², for free!

Cite As Get BibTex

Hana Dal Poz Kouřimská, André Lieutier, and Mathijs Wintraecken. A Free Lunch: Manifolds of Positive Reach Can Be Smoothed Without Decreasing the Reach. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 37:1-37:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.37

Author Details

Hana Dal Poz Kouřimská
  • University of Potsdam, Germany
André Lieutier
  • Aix-en-Provence, France
Mathijs Wintraecken
  • Inria Centre d'Université Côte d'Azur, Sophia Antipolis, France

Funding

  • Dal Poz Kouřimská, Hana: Supported by the DFG project No. 524578210.
  • Wintraecken, Mathijs: Supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411, the Austrian science fund (FWF) grant No. M-3073, the ANR grant StratMesh (ANR-24-CE48-1899), and the welcome package from IDEX of the Université Côte d'Azur, (ANR-15-IDEX-01).

Acknowledgements

We thank Jean-Daniel Boissonnat for discussion. We would also like to acknowledge the organizers of the workshop on "Algorithms for the Medial Axis", and Erin Chambers in particular, for giving an impulse to this research. We further thank Victor Bangert for encouragement.

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