Suppose that $(X,\rho)$ is a compact doubling metric space. 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.
The Assouad embedding theorem, see e.g. this paper for a recent formulation, 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.
As remarked in this old MO post, in this paper of Katz and Katz (with un unpublished quantitative version found here) shows we know that there is a bi-Lipschitz embedding of any closed and connected Riemannian manifold $(M,g)$ into some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.
I comment that smoothness is not needed in (1) since the existence of $\Phi$ is obvious for any finite metric space.
