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arXiv:2401.04034v5 Announce Type: replace
Abstract: We define a notion of Morse function and establish Morse theory-like theorems over offsets of any compact set in a Euclidean space at regular values of their distance function. Using non-smooth analysis and tools from geometric measure theory, we prove that the homotopy type of the sublevels sets of these Morse functions changes at a critical value by gluing exactly one cell around each critical point.
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