Suppose that $(X,\rho)$ is a compact [doubling metric space][1].  Does there necessarily exist an $\epsilon>0$ and a maximal $\epsilon$-net $\{x_i\}_{i=1}^n\subseteq X$ such that the map
$$
\begin{aligned}
\Phi:(X,\rho) & \rightarrow (\mathbb{R}^n,|\cdot|_2) \\
x&\mapsto \big(\rho(x,x_i)\big)_{i=1}^n
\end{aligned}
$$
is bi-Lipschitz?  (A trivial upper-Lipschitz bound of $\sqrt{n}$ is clear but the lower-Lipschitz bound is far from obvious for me).

---

My question is rooted in the following observations.


**Motivation/Intuition:**
The motivation for my question is rooted in the following two observations.

 1. The Assouad embedding theorem, see e.g. [this paper for a recent formulation][2], shows that every doubling metric space admits a bi-Hölder embedding into a Euclidean space.  Moreover, it is known that bi-Hölder is necessary, due to the *global* non-embeddability of the Heisenberg group, since the distortion of any closed ball diverges as the radius grows; [this paper][3].  

 2. As remarked in this [old MO post][4], in [this paper of Katz and Katz][5] (with un unpublished quantitative version [found here][6]) shows we know that there is a bi-Lipschitz embedding of any closed and connected Riemannian manifold $(M,g)$ into some Euclidean space $(\mathbb{R}^n,|\cdot|_2)$ given by
$$
\varphi:\,M\ni x\mapsto \big(\rho_g(x,x_i)\big)_{i=1}^n \in \mathbb{R}^n
$$
where $\{x_i\}_{i=1}^n$ is any maximal $\epsilon$-net for some sufficiently small $\epsilon>0$ and $\rho_g$ is the geodesic distance on $(M,g)$.
Clearly, compactness is needed here, since it is well-known that the hyperbolic plane cannot be bi-Lipschitz embedded into any Euclidean space.

 3. I comment that smoothness is not needed in (1) since the existence of $\Phi$ is obvious for any finite metric space.

----
**Update:**

**Claim:** if there exists some $0<\epsilon\le 1$ and a finite set $Y\subseteq X$ (depending on $\epsilon>0$) with the property that for every $x,z\in X$ there is some (possibly not unique) $y_{x,z}\in Y$ satisfying 
$$
\epsilon \rho(x,z)\le |\rho(x,y_{x,z})-\rho(z,y_{x,z})|.
$$
If this case, the finiteness of $Y$ implies that
$$
\epsilon \rho(x,z)\le |\rho(x,y_{x,z})-\rho(z,y_{x,z})|
\le \max_{y\in Y}\, |\rho(x,y)-\rho(z,y)|
\le \|\Phi(x)-\Phi(y)\|_2
\le |Y|^{1/2}\,\max_{y\in Y}\, |\rho(x,y)-\rho(z,y)| \le |Y|^{1/2}\rho(x,y)
$$ 
where I used the fact that $|Y|<\infty$ and $x\mapsto \rho(x,y)$ is $1$-Lipschitz for every $y\in Y$.


  [1]: https://en.wikipedia.org/wiki/Doubling_space
  [2]: https://web.math.princeton.edu/~naor/homepage%20files/assouad-N(K).pdf
  [3]: https://web.math.princeton.edu/~naor/homepage%20files/UCHeis9.pdf
  [4]: https://mathoverflow.net/questions/417185/finite-approximations-to-the-kuratowski-fr%C3%A9chet-embedding
  [5]: https://link.springer.com/article/10.1007/s10711-010-9497-4
  [6]: https://arxiv.org/pdf/1305.1529.pdf