Is there an example of a metric space $X$ with two points $p$ and $q$ so that for every $r>0$ the ball with radius $r$ and center $p$ is isometric to the ball with radius $r$ and center $q$ and yet no automorphism of $X$ sends $p$ to $q$?
2026-03-28 14:05:36.1774706736
Metric space with two similar points which are not in the same orbit.
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Yes, such spaces exist. See this MathOverflow answer for two nice examples: https://mathoverflow.net/questions/182719/if-all-balls-at-x-and-y-are-isometric-is-there-an-isometry-sending-x-to-y