Given a closed set S in Euclidean space, one can consider the (generalized) gradient of the distance function to S.
We are interested in the study of this gradient, and especially in the (generalized) critical points Z(S) of the distance function, which play an important role in computational geometry, and in particular in the study of the Cech complex of S.
In general, the set of critical points Z(S) can be poorly behaved; however, we show that Z(S) becomes very regular and stable when S=M is a generically embedded submanifold.
More precisely, for any compact abstract manifold M, the set of embeddings of M into a Euclidean space such that their image satisfies our strong regularity and stability conditions is open and dense in the Whitney C2-topology.
When those conditions are satisfied, the distance function to the embedded submanifold is a topological Morse function and the persistent homology of the submanifold is very regular with respect to subsamplings; this has consequences in topological data analysis.
