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

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.

SUBMITTER: Boissonnat JD

PROVIDER: S-EPMC6777504 | biostudies-literature | 2019

REPOSITORIES: biostudies-literature

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

Boissonnat Jean-Daniel JD Lieutier André A Wintraecken Mathijs M

Journal of applied and computational topology 20190724 1

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 i ...[more]

PMID: 31633006

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

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

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