Fonction distance et singularités. (French) Zbl 0547.58011

Let Y be a closed subset of \({\mathbb{R}}^ n\) endowed with the Euclidean metric d. Define the function \(f:{\mathbb{R}}^ n\to {\mathbb{R}}\) as \(f(x)=d^ 2(x,Y)\). Also for each k (k a positive integer, \(k=\infty\) or \(k=\omega)\) define the set \(reg_ kf\) as consisting of those \(x\in {\mathbb{R}}^ n\) in a neighbourhood of which f is of class \(C^ k\), and the set \(\sin g_ kf\) as its complement in \({\mathbb{R}}^ n\). Similarly, let \(reg_ kY\) be the set of those \(y\in Y\) in a neighbourhood of which Y is a submanifold of \({\mathbb{R}}^ n\) of class \(C^ k\). Again, \(\sin g_ kY\) is the complement of \(reg_ kY\) in \({\mathbb{R}}^ n\). The paper is concerned with commenting on and proving the relation \(\sin g_ kY=Y\cap\sin g_ kf\) for any \(k\geq 2\). (The example of \(Y=(-\infty,0]\subset {\mathbb{R}}\) shows that for \(k=1\) this is false.)
Reviewer: J.Virsik


58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
53B25 Local submanifolds
32B20 Semi-analytic sets, subanalytic sets, and generalizations