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

Jean-Daniel Boissonnat 1 André Lieutier 2 Mathijs Wintraecken 1 
1 DATASHAPE - Understanding the Shape of Data
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
Abstract : In this paper we discuss three results. The first two concern general sets of positive reach: We first characterize the reach by means of a bound on the metric distortion between the distance in the ambient Euclidean space and the set of positive reach. 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 distance between the points and the reach.
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Submitted on : Tuesday, December 12, 2017 - 2:47:26 PM
Last modification on : Friday, February 4, 2022 - 3:12:13 AM


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  • HAL Id : hal-01661227, version 1


Jean-Daniel Boissonnat, André Lieutier, Mathijs Wintraecken. The reach, metric distortion, geodesic convexity and the variation of tangent spaces. 2017. ⟨hal-01661227⟩



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