Critical points of the distance function to a generic submanifold

Abstract

In general, the critical points of the distance function $d_{\mathsf{M}}$ to a compact submanifold $\mathsf{M} \subset \mathbb{R}^D$ can be poorly behaved. In this article, we show that this is generically not the case by listing regularity conditions on the critical and $\mu$-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 $i:M\rightarrow \mathbb{R}^D$ 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 distance function to $i(M)$ satisfies Morse-like conditions and that the critical points of the distance function to an $\varepsilon$-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 allows for pathological situations.