TITLE: Given an example of a metric space in which every sphere has two centers
QUESTION [7 upvotes]: 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 :(

REPLY [2 votes]: 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 $-x$.