Bi-Lipschitz embeddings: Hilbert spaces vs. finite-dimensional Euclidean spaces

Let $(X,d)$ be a metric space. Say it is doubling if there is a finite constant $K$ such that every ball $B(x,r)$, for $x\in X$ and $r>0$, can be covered by at most $K$ balls of radius $r/2$. Suppose that $X$ is doubling and that there are a Hilbert space $H$, a map $f:X\to H$, and constants $0<a\leq b<\infty$ such that, for every $x,y\in X$,

$$a\,d(x,y)\leq \|f(x)-f(y)\|_H\leq b\,d(x,y).$$

Must there exist a finite positive integer $N$, a map $g:X\to\mathbb{R}^N$, and constants $0<a'\leq b'<\infty$ such that, for every $x,y\in X$,

$$a'\,d(x,y)\leq \|g(x)-g(y)\|_2\leq b'\,d(x,y)?$$

The question asks for some finite dimension and finite distortion; it does not prescribe uniform bounds on either across all such spaces. AimPL, Problem 2.4 Assouad’s theorem with dimension independent of the snowflaking

Background

This question is also known as the Lang–Plaut problem. Its significance is that doubling controls the number of smaller balls needed to cover a ball, while Hilbert embeddability supplies a different, global constraint on distances. Neither condition alone gives the desired conclusion: doubling spaces need not embed bi-Lipschitzly into finite-dimensional Euclidean space, and the problem asks whether the two conditions together suffice. Assouad’s theorem with dimension independent of the snowflaking

Assouad’s theorem gives a nearby but weaker conclusion: for a doubling space and each snowflaking parameter in the stated range, the snowflaked metric embeds bi-Lipschitzly into finite-dimensional Euclidean space. The dimension and distortion in that result may depend on the parameter. Since snowflaking changes the metric, this does not itself answer whether the original metric embeds. The Lang–Plaut formulation recasts the question quantitatively: for each doubling constant, can one bound both the Euclidean dimension and distortion for all doubling subsets of Hilbert space with that constant? Assouad’s theorem with dimension independent of the snowflaking