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Manifolds of Positive Reach, Differentiability, Tangent Variation, and Attaining the Reach

Authors André Lieutier , Mathijs Wintraecken

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LIPIcs.SoCG.2026.74.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Reach
  • Manifolds
  • Differentiability class
  • Lipschitz continuity
  • Tangent space

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Abstract

This paper contains three main results. 
Firstly, we give an elementary proof of the following statement: Let ℳ be a topological manifold without boundary embedded in R^d. If ℳ has positive reach, then ℳ can locally be written as the graph of a C^{1,1} function from the tangent space to the normal space. Conversely if ℳ can locally be written as the graph of a C^{1,1} function from the tangent space to the normal space, then ℳ has positive reach. The result was hinted at by Federer when he introduced the reach, and proved by Lytchak. Lytchak’s proof relies heavily on CAT(k)-theory. The proof presented here uses only basic results on homology.
Secondly, we give optimal Lipschitz-constants for the derivative, in other words we give an optimal bound for the angle between tangent spaces in term of the distance between the points. We stress that Lytchak did not provide any bound, let alone an optimal one, making his proof, although interesting from a mathematical perspective, ineffectual in an algorithmic setting. To provide precise and optimal bounds on the angle between tangent spaces, we formally introduce the local reach for sets of positive reach, based on Aamari et al.’s discussion for C² manifolds. We prove that the local reach of a manifold is completely characterized by the variation of tangent spaces. This improves earlier results, that were either suboptimal or assumed that the manifold was C². 
Thirdly, we show that the value of the reach is equals minimum of the local reach of the set and a global bottleneck for any set. This generalizes a result by Aamari et al. which explains how the reach is attained for C² manifolds.

Cite As Get BibTex

André Lieutier and Mathijs Wintraecken. Manifolds of Positive Reach, Differentiability, Tangent Variation, and Attaining the Reach. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 74:1-74:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.74

Author Details

André Lieutier
  • Aix-en-Provence, France
Mathijs Wintraecken
  • Inria Centre d'Université Côte d'Azur, Sophia Antipolis, France

Funding

  • 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 the reviewers for their comments that improved the exposition.

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