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I'm looking for such literature as exists relevant to the following problem.

Problem Given a compact, path-connected region $E$ on the plane and a positive constant $r$. Find (if possible) a path $P$ in $E$ with the following property: Given a point $p \in E$, there exists a point $q $ lying on $P$ such that $||q-p|| \leq r$ , where $||$ is Euclidean distance.

I am also interested in the following variants of the Problem:

  1. What conditions have $E$ has to satisfy for a $P$ to exist?
  2. Can we find the set of all such paths?
  3. If we restrict $P$ to be $n$-times differentiable, under what conditions does it exist?

I'm aware that these variants may lead to a combinatorial explosion of answers for each special case. Hence, would be grateful for links to foundational literature.

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$P$ always exists---take a countable dense subset of $E$, enumerate it, and connect the points in order... – Daniel Litt Mar 24 '11 at 7:06
    
@Daniel: How do we connect points when $P$ has to be $n$-times differentiable? – Ganesh Mar 24 '11 at 7:15
    
What do you mean when you say "region"? – S. Carnahan Mar 24 '11 at 7:58
    
The Hahn–Mazurkiewicz theorem says that, under mild topological assumptions on $E$, there exists a surjective path (see en.wikipedia.org/wiki/Space-filling_curve). Of course these space-fillers may be extremely non-differentiable. – Mark Grant Mar 24 '11 at 10:07
    
I'm assuming that region implies open (otherwise it's easy to create problems). In that case, Daniel Litt's solution sounds pretty solid to me. Making the path n times differentiable should not be a problem. – Thierry Zell Mar 24 '11 at 11:10

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