Given two metric spaces $(X,d_X), (Y, d_Y)$, a bi-Lipschitz map $f:X \to Y$ and a finite set of points $\{x_1, \ldots, x_n\} \in X$. Consider in addition, that $X$ is a vector space over $\mathbb{R}$, concretely some $l^p$ space. Under which conditions is it possible to find points $\{y_1, \ldots, y_n\} \in Y$ such that for each pair of points $x_i,x_j$ we have $d_X(x_i, x_j) = d_Y(y_i, y_j)$?

This can go very wrong in totally disconnected spaces.

For instance, if $X$ and $Y$ are both finite, one can adjust the distances between points via the bi-Lipschitz map so that every collection of points gets their distances changed.

Even further, you can let $X$ be the rationals and let $Y$ be constructed by recursively moving each point of $X$ to a point in a distinct coset of the rationals, moving each point by only $1/2^n$, say. This is another example of a pair of spaces where you can never map a collection of points isometricly.

let $X,Y$ be bilipschitz homeomorphic metric spaces. Under what condition is it true that $X$ and $Y$ are "finitely isometric"?(where "finitely isometric" means that every finite subset of $X$ is isometric to some finite subset of $Y$?). The question sounds a bit vague, what do you except other than trivial restatements or a list of spaces that are both bilipschitz and finitely isometric? $\endgroup$ – YCor Dec 18 '15 at 14:354more comments