Suppose $A$ and $B$ are disjoint subsets of the plane, both closed, nonempty, and connected. Define $E(A, B)$ as the set of points in the plane equidistant from $A$ and $B$. For example, if $A$ is a point and $B$ is a straight line, $E$ is a parabola.
(1) I think that $E$ is always homeomorphic to a circle or a line. Is that right?
(2) Are there any generalizations? For example, if instead of the plane we take the ambient space to be any $n$-manifold with a metric, is $E$ something like an $(n - 1)$-dimensional CW complex?
And here are some bonus questions...
Are there conditions on $A$ and $B$ which will guarantee that $E$ is a submanifold of codimension 1? For example, in $\mathbb{R}^3$, $E$ is not always a surface even if $A$ and $B$ are connected. (To see this, let $A$ and $B$ be like two forks kissing: for example, $A$ is the union of the three line segments given by the sequence $(-1, 0, 0)$, $(-1, 0, 4)$, $(1, 0, 4)$, $(1, 0, 0)$, and $B$ is given similarly by $(0, 1, 4)$, $(0, 1, 0)$, $(0, -1, 0)$, $(0, -1, 4)$.) But perhaps $E$ is a surface if $A$ and $B$ are separated by a hyperplane.
What do we get in $\mathbb{R}^n$ if $A$ and $B$ are finite?
I think I can prove that if $A$ and $B$ are graphs of continuous functions $\mathbb{R} \rightarrow \mathbb{R}$ , then $E$ is too. Is there a similar result for $C^k$ functions? Is there a nice description of the $E$ function if the $A$ and $B$ functions are, say, polynomials?