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Let $A \subset X$ and $B \subset X$ be two isometric subsets of a metric space $X$. So there is an isometry $f: A \to B$. Say that a metric space $X$ has the superposition property (my terminology) ...

This is a follow-up to the question of Joseph O'Rourke Which metric spaces have this superposition property? A metric space $X$ will be called all-set-homogeneous if for any subset $A\subset X$ any ...