The reach of subsets of manifolds Boissonnat, Jean Daniel Wintraecken, Mathijs ; https://orcid.org/0000-0002-7472-2220 Kleinjohann (Archiv der Mathematik 35(1):574–582, 1980; Mathematische Zeitschrift 176(3), 327–344, 1981) and Bangert (Archiv der Mathematik 38(1):54–57, 1982) extended the reach rch(S) from subsets S of Euclidean space to the reach rchM(S) of subsets S of Riemannian manifolds M, where M is smooth (we’ll assume at least C3). 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−1p(S)) at q, assuming that rchM(S) and the geodesic distance dM(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. Springer Nature 2023 info:eu-repo/semantics/article doc-type:article text http://purl.org/coar/resource_type/c_2df8fbb1 https://research-explorer.ista.ac.at/record/12763 Boissonnat JD, Wintraecken M. The reach of subsets of manifolds. <i>Journal of Applied and Computational Topology</i>. 2023;7:619-641. doi:<a href="https://doi.org/10.1007/s41468-023-00116-x">10.1007/s41468-023-00116-x</a> eng info:eu-repo/semantics/altIdentifier/doi/10.1007/s41468-023-00116-x info:eu-repo/semantics/altIdentifier/issn/2367-1726 info:eu-repo/semantics/altIdentifier/e-issn/2367-1734 info:eu-repo/semantics/openAccess