A Morse-Sard theorem for the distance function on Riemannian manifolds (Q1434141)

The author establishes the following result: if \(N\) is a closed submanifold of dimension \(d\) and of class \(C^k\) in \(\mathbb R^n\) with \(k>2d(n+1)\), then the set of critical values of the distance function from \(N\) is of Lebesgue measure zero. Here a critical point \(u\) is considered in the sense that \(0\) belongs to Clarke's generalized gradient at \(u\).