Does the cut locus of a submanifold have Lebesgue measure zero?

Question

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, a positive answer is given when $W$ is of class $C^r$ with $r\geq 3$:

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

Answer 1

I read the paper of Mantegazza and Menucci too quickly: in the discussion before Remark 3.8 a counterexample for $r=1$ is given.