# Sobolev norm of distance function on Riemannian manifold

$$\DeclareMathOperator\SL{SL}$$Suppose $$M$$ is a Riemannian manifold with distance function $$d:M\times M \rightarrow [0,\infty)$$. If it helps let $$M$$ be a Lie group with finite Haar measure $$\mu$$ and left invariant metric (like $$\SL(n,\mathbb Z)\backslash \SL(n,\mathbb R)$$). Then for fixed $$q$$ the distance function $$d(q,.)$$ is smooth at a neighborhood of $$q$$ at points $$p\neq q$$.

But what can you say if the point $$q$$ is replaced by a set $$V\subseteq M$$. Are there any sufficient conditions on $$V$$ such that $$d(V,.)$$ is smooth? What about if $$V$$ is open? Can you establish smoothness at the boundary $$\partial V$$ if you look at $$d(V,.)^2$$ instead? Are there any estimates for the $$(2,l)$$-th sobolev norm for $$d(V,.)$$?

Thanks very much!

You should definitely have a look at this paper of Mantegazza-Mennucci, which gives you quite a lot of information on how the regularity of the set $$V$$ influences the regularity of $$d(V,\cdot)$$ and $$d(V,\cdot)^2$$.
For example, in Proposition 4.2 they show that if $$V$$ is a $$C^r$$ submanifold of $$M$$ (of any dimension) with $$r\geq 2$$, then there exists an open set $$\Omega\supset V$$ such that $$d(V,\cdot)$$ is $$C^r$$ in $$\Omega\backslash V$$ and $$d(V,\cdot)^2$$ is $$C^r$$ in $$\Omega$$.