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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 infin …
Bjørn Kjos-Hanssen's user avatar