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As the title says. Let $A^n$ be an $n$-dimensional closed Alexandrov space. Does it admit a bi-Lipschitz embedding into Euclidean space $\mathbb R^N$ for sufficiently large $N$?

I know there are some spaces that do not admit such an embedding; for example, a theorem by Pansu says that:

The Heisenberg group equipped with the Carnot-Caratheodory distance does not biLipschitz embed into $\mathbb R^n$, for any $n$.

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    $\begingroup$ The Heisenberg group doesn't embed isometrically with respect to Euclidean distance, but Enrico Le Donne proved that there are Nash-type embeddings that preserve the length of every curve! See arxiv.org/abs/1005.1623 . $\endgroup$ – Robert Young Apr 12 '13 at 16:04
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See Distance embedding (27.5) in our book

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  • $\begingroup$ Dear Anton, are chapters 13 and 15 available? $\endgroup$ – Suvrit Apr 12 '13 at 23:14
  • $\begingroup$ @Suvrit, everything which is nearly ready is there, check it time to time, maybe more sections will appear. If you really-really need some sections I could send it with a "read on your own risk" notice --- send me an e-mail. $\endgroup$ – Anton Petrunin Apr 13 '13 at 18:23
  • $\begingroup$ Thanks Anton. I'll email if you if this need arises (I'm mostly interested in the theory of first-order approximation in metric spaces, and in particular in CAT(0) spaces and slight perturbations of these spaces. $\endgroup$ – Suvrit Apr 13 '13 at 22:00
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In my preprint "Bi-Lipschitz embeddings of SRA-free spaces into Euclidean spaces" (https://arxiv.org/abs/1906.02477) I prove the quantitative version of this statement.

Theorem 2. For $n \in \mathbb{N}$, $k < 0$ and $R > 0$ there exist $D > 0$ and $N \in \mathbb{N}$ satisfying the following. For every $n$-dimensional Alexandrov space of curvature $\ge k$ and every $x \in X$ there exists an embedding $\phi:B_R(x) \rightarrow \mathbb{N}^N$ which bi-Lipschitz distortion does not exceed $D$.

For $n \in N$ there exist $D_0 > 0$ and $N_0 \in \mathbb{N}$ such that every $n$-dimensional Alexandrov space of non-negative curvature allows an embedding into $\mathbb{N}^{N_0}$ which bi-Lipschitz distortion does not exceed $D_0$.

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