TITLE: Which metric spaces have this superposition property?
QUESTION [9 upvotes]: 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) if, for every pair of isometric subsets $A$, $B$, there is an isometry of $X$, $F: X \to X$, that superimposes $A$ onto $B$:  $F(A) = B$, i.e. $F$ places $A$ on top of $B$.

Which metric spaces have this superposition property?

Euclidean space $\mathbb{E}^d$ has this property.
But it seems the punctured plane does not: e.g. if $A$ is the point $(1,0)$ and $B$ is the point $(-2,0)$, then (I believe) there is not an isometry of the whole punctured plane that maps $A$ onto $B$.
Has this property been studied before?  If so, under what name?
I am (clearly) unschooled in this area. Thanks for pointers and/or examples!

REPLY [6 votes]: If you restrict sets A, B to be simply points, then you are asking for spaces which have the property that for all points A,B\in X there exists an isometry T from X onto X so that T(A)=B (am I understanding you correctly?). Such spaces are called transitive. It is known that if a finite dimensional space is transitive then it is isometric to a Euclidean space, I think that this is a result of Mazur. It is an old, still open problem, whether every separable transitive Banach space is isometric to a Euclidean space. This problem is called Banach-Mazur problem and it goes back to 1930's. There has been a lot of work on this problem and it is connected to other interesting problems.
If you require that every 2 points can be mapped by a surjective isometry onto any other 2 points with the same distance, then I think the space is called 2-transitive. Similarly one defines n-transitive. I believe, but I am not 100% certain that 2-transitive Banach spaces have to be isometric to a Euclidean space. You might check work of V. Mascioni.