Suppose there is a compact metric space $(X,\rho)$ and a Euclidean space $\mathbb{R}^n$. 


There is a sequence of unequal points $\{x\_1,...x\_N\}$ in $X$ such that all metrics $\rho(x\_i,x\_j)$ are known and $f(x\_i)=a\_i$ for some $a\_i$ in $\mathbb{R}^n$ whereas:


$$ \forall x\_i,x\_j \in \{x\_1,...x\_N\} . ||a\_i-a\_j|| \leq L \cdot \rho(x\_i, x\_j) $$


for a fixed $L$.


Suppose also that $\{x\_1,...x\_N\}$ form a finite cover of $X$ by balls of some suitable (known) radius.


If $X$ were just a compact interval, $\{x\_1,...x\_N\}$ its partition, and $n$ were $1$, we could certainly construct a piecewise-linear function $f:X \rightarrow \mathbb{R}^n$ such that:


$$ f(x\_i)=a\_i \\ \forall x,y \in X . ||f(x)-f(y)|| \leq L \cdot \rho(x, y) $$ 


with a slope controlled by $L$.


It should be also possible in certain cases if $X \subset \mathbb{R}^m$.


**Is it possible to construct such a function for a general compact metric space? And if $X$ is a subset of a Euclidean space?**



Constructing a function over a metric space through given points
Applying rescaling you can assume that $L=1$.


We look for a 1-Lipschitz piecewise linear maps $f\colon\mathbb{R}^m\to\mathbb{R}^n$. 


If $m=n$ we get $f$ from Brehm's theorem;
it says that there is a piecewise distance preserving map of that type (in particular 1-Lipschitz and piecewise linear). 
See Brehm, U., *Extensions of distance reducing mappings to piecewise
congruent mappings on $\mathbb{R}^m$* and also [our paper](http://arxiv.org/abs/1405.6606) written for kids.


If $m<n$ we can think that $\mathbb{R}^m$ is a subspace of $\mathbb{R}^n$; in this case apply Brehm's theorem and restrict the obtained map to $\mathbb{R}^m$. 


If $m>n$ we can think that $\mathbb{R}^n$ is a subspace of $\mathbb{R}^m$; in this case apply Brehm's theorem and compose the obtained map with the projection to $\mathbb{R}^n$.
Since projection linear and 1-Lipschitz, so it the composition.


