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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?

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    Gosh...not even the ghost of an answer. Do I really need to put a bounty on this one? Surely someone knows something.2012-10-16
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    I think you want to think of $E$ as the zero level set of $e(x) = d(x,A) - d(x,B)$. Then I imagine you might able to prove that the sets $\{x: e(x) < 0\}$ and $\{x: e(x) > 0\}$ are connected, from which nice things should follow.2012-10-20
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    I have to say I don't understand your example in $\mathbb R^3$. When $i=1$, $A_1 = e^{2\pi/4} \not\in S^1$. Perhaps you meant $A_k = e^{2\pi ik/4}$ instead? And how do the four points $A_0A_0'A_2A_2'$ (which I presume are $(1,0,0)$, $(1,0,4)$, $(-1,0,0)$, $(-1,0,4)$ in $\mathbb R^3$) specify a line?2012-10-20
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    Thanks @RahulNarain, I fixed up that example.2012-10-22

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