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Geodesics of Length Less Than πR in a Set of Reach R Are Unique and Continuous with Respect to the Endpoints

Authors André Lieutier , Mathijs Wintraecken

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LIPIcs.SoCG.2026.75.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Reach
  • geodesics
  • metric geometry

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Abstract

Positive reach underpins many results in computational geometry and topology. It is used for triangulation criteria, topological inference, and manifold learning. The geometric properties of these sets have therefore been studied intensely. Here we focus on the shortest paths or minimizing geodesics in these sets. Our main result states that minimizing geodesics of length strictly less than π R in a set of reach R are unique. This in turn implies that such minimizing geodesics are continuous with respect to the endpoints.

Cite As Get BibTex

André Lieutier and Mathijs Wintraecken. Geodesics of Length Less Than πR in a Set of Reach R Are Unique and Continuous with Respect to the Endpoints. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 75:1-75:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.75

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) M-3073. The welcome package from IDEX of the Université Côte d'Azur, ANR-15-IDEX-01. The French National Research Agency (ANR) under grant StratMesh, ANR-24-CE48-1899.

Acknowledgements

We thank Jean-Daniel Boissonnat for discussion and Eddie Aamari for his insight in possible applications. We are indebted to the reviewers for many suggestion and in particular noting that Corollary 10 immediately follows from the results in our submission.

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