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.

Highlights

  • Metric distortion quantifies the maximum ratio between geodesic and Euclidean distances for pairs of points in a set S

  • In the first part of this paper, we provide tight bounds on metric distortion for sets of positive reach and, in a second part, we consider submanifolds of Rd and bound the angle between tangent spaces at different points

  • The metric distortion and tangent variation bounds for C1,1 manifolds presented in this paper suffice to extend the triangulation result of C2 manifolds embedded in Euclidean space given in Boissonnat et al (2018) to arbitrary manifolds with positive reach, albeit with slightly worse constants

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Summary

Introduction

Metric distortion quantifies the maximum ratio between geodesic and Euclidean distances for pairs of points in a set S. The metric distortion and tangent variation bounds for C1,1 manifolds presented in this paper suffice to extend the triangulation result of C2 manifolds embedded in Euclidean space given in Boissonnat et al (2018) to arbitrary manifolds with positive reach, albeit with slightly worse constants. Based on our new characterization of the reach by metric distortion, we can prove that the intersection of a set of positive reach with a ball with radius less than the reach is geodesically convex This result is a far reaching extension of a result of Boissonnat and Oudot (2003) that has attracted significant attention, stating that, for smooth surfaces, the intersection is a pseudo-ball. In the final section we reproof some of the results of the first section using differential geometrical techniques

Metric distortion and geodesic convexity
Projection of the middle point
Metric distortion
Bounds for C2 submanifolds
A topological result
From manifold to tangent space and back
The angle bound This section revolves around the following observation
Metric distortion and geodesic convexity for C2 submanifolds
Convexity
Conclusions and future research
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