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 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'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.