%AKim, Jisu%AShin, Jaehyeok%AChazal, Frédéric%ARinaldo, Alessandro%AWasserman, Larry%BJournal Name: Leibniz international proceedings in informatics; Journal Volume: 164; Journal Issue: 36th International Symposium on Computational Geometry (SoCG 2020) %D2020%I %JJournal Name: Leibniz international proceedings in informatics; Journal Volume: 164; Journal Issue: 36th International Symposium on Computational Geometry (SoCG 2020) %K %MOSTI ID: 10175450 %PMedium: X %THomotopy Reconstruction via the Cech Complex and the Vietoris-Rips Complex %XWe derive conditions under which the reconstruction of a target space is topologically correct via the Čech complex or the Vietoris-Rips complex obtained from possibly noisy point cloud data. We provide two novel theoretical results. First, we describe sufficient conditions under which any non-empty intersection of finitely many Euclidean balls intersected with a positive reach set is contractible, so that the Nerve theorem applies for the restricted Čech complex. Second, we demonstrate the homotopy equivalence of a positive μ-reach set and its offsets. Applying these results to the restricted Čech complex and using the interleaving relations with the Čech complex (or the Vietoris-Rips complex), we formulate conditions guaranteeing that the target space is homotopy equivalent to the Čech complex (or the Vietoris-Rips complex), in terms of the μ-reach. Our results sharpen existing results. %0Journal Article