MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
up vote 3 down vote accepted

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 $\Omega$.

This paper also contains many useful references, like this one, which also discusses Sobolev norms.

share|cite|improve this answer
    
Thank you so much, this looks very promising – Florek Sep 8 '12 at 3:19

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.