Let $M$ be a Riemannian manifold where closed balls are compact. Let $W\subset M$ be a submanifold of $M$ that is closed as a subset of $M$. Define the *cut locus* of $W$ in $M$ as 
$$\mathrm{Cut}(W;M)=\overline{\{x\in M\mid \exists y,z\in W,\text{with $y\neq z$ and } d(x,W) = d(x,y) = d(x,z)\}}$$
where the line denotes "closure".

> Does $\mathrm{Cut}(W;\mathbb R^n)$ have Lebesgue measure zero?

> What happens for an arbitrary $M$?



What I have found:

 - The Hopf-Rinow theorem implies that when $W$ is a point the answer is yes.
 - In Corollary 4.12 of [Mantegazza and Mennucci, Hamilton-Jacobi equations and distance functions on Riemannian manifolds,  Appl. Math. Optim. 47 (2003), no. 1, 1–25][1], a positive answer is given when $W$ is of class $C^r$ with $r\geq 3$:[![enter image description here][2]][2] [![enter image description here][3]][3]

What happens when $r<3$? Is there any counterexample in these cases?


  [1]: https://mathscinet-ams-org.sire.ub.edu/mathscinet-getitem?mr=1941909
  [2]: https://i.sstatic.net/8LiAv.png
  [3]: https://i.sstatic.net/lHmah.png