# The reach, metric distortion, geodesic convexity and the variation of tangent spaces

Boissonnat J-D, Lieutier A, Wintraecken M. 2019. The reach, metric distortion, geodesic convexity and the variation of tangent spaces. Journal of Applied and Computational Topology. 3(1–2), 29–58.

2019_JournAppliedComputTopol_Boissonnat.pdf 2.22 MB

Journal Article | Published | English
Author
Boissonnat, Jean-Daniel; Lieutier, André; Wintraecken, MathijsISTA
Department
Abstract
In this paper we discuss three results. The first two concern general sets of positive reach: we first characterize the reach of a closed set by means of a bound on the metric distortion between the distance measured in the ambient Euclidean space and the shortest path distance measured in the set. Secondly, we prove that the intersection of a ball with radius less than the reach with the set is geodesically convex, meaning that the shortest path between any two points in the intersection lies itself in the intersection. For our third result we focus on manifolds with positive reach and give a bound on the angle between tangent spaces at two different points in terms of the reach and the distance between the two points.
Publishing Year
Date Published
2019-06-01
Journal Title
Journal of Applied and Computational Topology
Volume
3
Issue
1-2
Page
29–58
ISSN
eISSN
IST-REx-ID

### Cite this

Boissonnat J-D, Lieutier A, Wintraecken M. The reach, metric distortion, geodesic convexity and the variation of tangent spaces. Journal of Applied and Computational Topology. 2019;3(1-2):29–58. doi:10.1007/s41468-019-00029-8
Boissonnat, J.-D., Lieutier, A., & Wintraecken, M. (2019). The reach, metric distortion, geodesic convexity and the variation of tangent spaces. Journal of Applied and Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-019-00029-8
Boissonnat, Jean-Daniel, André Lieutier, and Mathijs Wintraecken. “The Reach, Metric Distortion, Geodesic Convexity and the Variation of Tangent Spaces.” Journal of Applied and Computational Topology. Springer Nature, 2019. https://doi.org/10.1007/s41468-019-00029-8.
J.-D. Boissonnat, A. Lieutier, and M. Wintraecken, “The reach, metric distortion, geodesic convexity and the variation of tangent spaces,” Journal of Applied and Computational Topology, vol. 3, no. 1–2. Springer Nature, pp. 29–58, 2019.
Boissonnat J-D, Lieutier A, Wintraecken M. 2019. The reach, metric distortion, geodesic convexity and the variation of tangent spaces. Journal of Applied and Computational Topology. 3(1–2), 29–58.
Boissonnat, Jean-Daniel, et al. “The Reach, Metric Distortion, Geodesic Convexity and the Variation of Tangent Spaces.” Journal of Applied and Computational Topology, vol. 3, no. 1–2, Springer Nature, 2019, pp. 29–58, doi:10.1007/s41468-019-00029-8.
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