Critical points of the distance function to a generic submanifold
Résumé
In general, the critical points of the distance function d_M to a compact submanifold M ⊂ R^D can be poorly behaved. In this article, we show that this is generically not the case by listing regularity conditions on the critical and µ-critical points of a submanifold and by proving that they are generically satisfied and stable with respect to small C^2 perturbations. More specifically, for any compact abstract manifold M , the set of embeddings i : M → R^D such that the submanifold i(M) satisfies those conditions is open and dense in the Whitney C^2-topology. When those regularity conditions are fulfilled, we prove that the critical points of the distance function to an ε-dense subset of the submanifold (e.g. obtained via some sampling process) are well-behaved. We also provide many examples that showcase how the absence of these conditions can result in pathological cases.
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