2026-07-20T11:45:54Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:25843 2026-06-23T12:59:23Z ddc:004 open_access

A Free Lunch: Manifolds of Positive Reach Can Be Smoothed Without Decreasing the Reach Dal Poz Kouřimská, Hana Lieutier, André Wintraecken, Mathijs Reach Manifolds Smoothing Differentiability Differential topology Assumptions on the reach are crucial for ensuring the correctness of many geometric and topological algorithms, including triangulation, manifold reconstruction and learning, homotopy reconstruction, and methods for estimating curvature or reach. However, these assumptions are often coupled with the requirement that the manifold be smooth, typically at least C². In this paper, we prove that any manifold with positive reach can be approximated arbitrarily well by a C^∞ manifold without significantly reducing the reach. More precisely, given a manifold with reach R, we construct a manifold that is ε-close to it in the C¹ sense (both the manifold and its tangent spaces are close), and has reach at least R-ε. The proof employs techniques from differential topology - partitions of unity and smoothing using convolution kernels. This result implies that nearly all theorems established for C² or manifolds with a certain reach naturally extend to manifolds with the same reach, even if they are not C², for free! Schloss Dagstuhl – Leibniz-Zentrum für Informatik Hana Dal Poz Kouřimská and André Lieutier and Mathijs Wintraecken 2026 Is Part Of LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2026.37 urn:nbn:de:0030-drops-258434 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.37 eng https://creativecommons.org/licenses/by/4.0/legalcode