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Journal Articles Journal of Applied and Computational Topology Year : 2023

The reach of subsets of manifolds

Abstract

Kleinjohann [1] and Bangert [2] extended the reach rch(S) from subsets S of Euclidean space to the reach rch M (S) of subsets S of Riemannian manifolds M, where M is smooth (we'll assume at least C^3). Bangert showed that sets of positive reach in Euclidean space and Riemannian manifolds are very similar. In this paper we introduce a slight variant of Kleinjohann's and Bangert's extension and quantify the similarity between sets of positive reach in Euclidean space and Riemannian manifolds in a new way: Given p ∈ M and q ∈ S, we bound the local feature size (a local version of the reach) of its lifting to the tangent space via the inverse exponential map (exp^{ −1}_p (S)) at q, assuming that rch M (S) and the geodesic distance d_ M (p, q) are bounded. These bounds are motivated by the importance of the reach and local feature size to manifold learning, topological inference, and triangulating manifolds and the fact that intrinsic approaches circumvent the curse of dimensionality.
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Dates and versions

hal-04083524 , version 1 (27-04-2023)

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Jean-Daniel Boissonnat, Mathijs Wintraecken. The reach of subsets of manifolds. Journal of Applied and Computational Topology, 2023, ⟨10.1007/s41468-023-00116-x⟩. ⟨hal-04083524⟩
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