The *Euclidean distortion* of a metric space $X$, denoted $c\_2(X)$, is the infimum of $c$ for which there exists a map $f\colon X\to\ell^2$ such that


$$d\_X(x,y) \leq \|f(x)-f(y)\|\_{\ell^2} \leq c\cdot d\_X(x,y).$$


In the introduction of their paper "[Nonembeddability theorems via Fourier analysis](https://arxiv.org/abs/math/0510547)", Khot and Naor mention the following in passing:



> 
> The Euclidean distortion of a metric space $X$ is relatively well
> understood: it is enough to understand the distortion of finite
> subsets of $X$, and for finite metrics there is a simple semidefinite
> program which computes their Euclidean distortion [39].
> 
> 
> 


Here, reference [39] is Linial, London, and Rabinovich's "[The geometry of graphs and some of its algorithmic applications](https://users.math.msu.edu/users/iwenmark/Teaching/CMSE890/EMBEDDING_The_Geometry_of_Graphs_and_Some_of_Its_Algorithmic.pdf)", which introduced the semidefinite program mentioned above. I'm interested in the first part of Khot and Naor's claim, which appears to suggest that


$$c\_2(X) = \sup\_{\substack{T\subseteq X \\ |T|<\infty}}c\_2(T)$$


for every metric space $X$. I.e., Euclidean distortion is *finitely determined*.


**Question:** Under what conditions on the metric space $X$ is $c\_2(X)$ finitely determined?


Here's what I know:


* The inequality $\geq$ holds since restricting a bilipschitz function $X\to\ell^2$ to $T$ can only improve the bilipschitz constants.
* The equality is known to hold if $X$ is locally finite. This follows from the main result of Ostrovskii's "[Embeddability of locally finite metric spaces into Banach
spaces is finitely determined](https://arxiv.org/abs/1103.0748)".
* The equality happens to hold for $X=\mathbb{R}/\mathbb{Z}$ with the quotient metric. This is a consequence of (the proof of) Claim 2.1 in Linial and Magen's "[Least-Distortion Euclidean Embeddings of Graphs: Products of Cycles and Expanders](https://core.ac.uk/download/pdf/82714902.pdf)", which establishes that $n$ equally spaced points in $X$ have distortion $\frac{n}{2}\sin\frac{\pi}{n}$. (This limits to the distortion of the obvious embedding of $X$ in $\mathbb{R}^2$.)
* The equality does not hold in pathological cases. For example, if $X$ is the power set of $\ell^2$ with trivial metric, then every finite subset of $X$ isometrically embeds into $\ell^2$ as the vertices of a regular simplex, but Cantor's theorem implies $c\_2(X)=\infty$.



When is Euclidean distortion finitely determined?
The problem is that you misdefined "Euclidean distortion". The infimum is over all maps into arbitrary Hilbert spaces, not just $\ell\_2$. Of course, you can use only $\ell\_2$ if the metric space is separable, but not when it is non separable. Now if every finite subset $F $of an infinite metric space $X$ embeds into $\ell\_2$ with constant $C+ |F|^{-1}$, then $X$ C-embeds into an ultra power of $\ell\_2$, and ultra powers of a Hilbert space are Hilbert spaces.


