Let $(X,d)$ be a metric space and $x,y \in X$. Assume that for all $r > 0$ the balls $B\_r(x)$ and $B\_r(y)$ are isometric.


Is it true that there exists an isometry of $X$ sending $x$ to $y$?



If all balls at $x$ and $y$ are isometric is there an isometry sending $x$ to $y$?
No. Let $x$ and $y$ be connected by an edge and let's use the graph distance as our metric.
At $x$, connect paths of length $n$ for each $n\in\mathbb N$. At $y$, do the same, but also connect an infinite path.


