Given an example of a metric space in which every sphere has two centers

Question

This is a question from Wilansky "Topology for analysis", P.15 Prob. 103

Maybe I was thinking too Euclidean, I can't come up some other "centers" of the sphere :(

Answer 1

I will answer under the assumption that you refer to sets $S_{\varepsilon}(x) = \{y: \, d(x,y) = \varepsilon\}$ as spheres, and $x$ as a center of the sphere.

On $\mathbb{Q}$, given any prime $p$, you can define the $p$-adic metric $|x-y|_p$ to be $p^{-n}$, where $n$ is the unique integer such that $x-y = p^n \frac{a}{b}$ with $a,b$ integers that are coprime to $p$. (If $x=y$, then define $|x-y|_p = 0$.)

This turns out to satisfy the ultrametric inequality $|x-y|_p \le \max\{|x|_p|, |y|_p\}$, and using this, you can show that, for any $x \in \mathbb{Q}$ and $\varepsilon > 0$, $S_{\varepsilon}(x) = S_{\varepsilon}(y)$ for any $y$ such that $|x-y|_p < \varepsilon$; in particular, there are many centers of any sphere.

Answer 2

+1 for using this book. It's one of my favorites.

To your question. Think about $S^1$ in $\Bbb R^2$. Then every 'sphere' is simply 2 points picked from $S^1$. This 'sphere' has 2 centers. The point that is circumferentially central to both points and the point diametrically opposed to that first point.

Think about given two points on $S^1$ and either of those 'centers' drawing a secant circle centered there and cutting through the two original points.

Answer 3

As I mentioned in my comment, the spheres $S^n$ provide a family of examples. A "sphere" in $S^{n}$ is a copy of $S^{n-1}$. If we normalize the "great circle distance" metric on $S^n$ to have $d(x,-x)=1$, then a sphere around $x$ of radius $r$ is also a sphere of radius $1-r$ around the antipodal point $-x$.