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.

Nonparametric estimation of directional highest density regions

“…Figure 1 in Jeong et al (2017) illustrates that ρ 1 (x, y) ≤ ρ 2 (x, y) for any pair of points x, y in the unit circle. Following Lemma 3 in Boissonnat et al (2019), a general upper bound for the ρ 2 (x, y) for all x, y ∈ S d−1 depending on ρ 1 (x, y). Specifically, it is possible to prove that ρ 2 (x, y) ≤ arcsin(ρ 1 (x, y)) for all x, y ∈ S d−1 when the constant r is equal to 1/2.…”

Triangulating Submanifolds: An Elementary and Quantified Version of Whitney’s Method

“…We first recall some notation. Similarly to [18], we let C (T p M , r 1 , r 2 ) denote the 'filled cylinder' given by all points that project orthogonally onto a ball of radius r 1 in T p M and whose distance to this ball is at most r 2 . We writeC…”

“…Proof Apart from Lemma 3.1, we will be using the following results from [18]: For a minimising geodesic γ on M with length parametrised by arc length, with γ (0) = p and γ ( ) = q, we have…”

“…which is an upper bound on the total number of faces of dimension less or equal to k that contain a given vertex. We have added because we want to have a safety margin if we have to consider the empty set (as will be apparent in (18)), and have a strict inequality. We now claim the following:…”

“…Manifolds, Tangent Spaces, Distances, and AnglesIn this section, we discuss some general results that will be of use. The manifold M ⊂ R d is a compact C 2 manifold with reach rch M .We adhere as much as possible to the same notation as used in[18]. The tangent bundle will be denoted by T M , while the tangent space at a point p is written as T p M .…”