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2. Embeddings

1. Deterministic proofs of bi-Lipschitz embedding theorems

Assouad’s embedding theorem asserts that to each doubling metric space $(X,d)$ and each $\epsilon \in (0,1)$, there corresponds $n$ so that the snowflake metric space $(X,d^\epsilon)$ admits a bi-Lipschitz embedding into ${\mathbb R}^n$. Naor–Neiman proved that $n$ can be chosen independent of the snowflake parameter $\epsilon$, for $\epsilon$ near one (say, $\epsilon>\tfrac12$). Their proof is probabilistic and nonconstructive. Can one give an explicit construction of such embedding?

Problem 2.1.

[Mario Bonk] Give a deterministic proof of Naor–Neiman’s improved Assouad embedding theorem.
Urs Lang remarks that a potential approach to this problem would combine the existence of bi-Lipschitz embeddings of doubling R-trees into finite-dimensional Euclidean spaces with a suitable adaptation of prior results on bi-Lipschitz embeddings of spaces of finite Nagata dimension into products of R-trees. For the former result, see Section 2.3 (and the references therein) in Lee-Naor-Peres, Trees and Markov convexity, GAFA 18 (2009). [In fact, this is a result about the vertex set of a graph-theoretical tree with weighted edges (= of different lengths), but since doubling trees are separable it is easy to promote this to a result about completely general R-trees.]
• Curvature conditions and bi-Lipschitz embeddings

Problem 2.3.

[Leonid Kovalev] Let $X$ be a doubling metric space. Can one formulate a curvature-type condition which, if satisfied by $X$, ensures that $X$ admits a bi-Lipschitz embedding into some finite-dimensional Euclidean space?
Such curvature condition should rule out all nonabelian Carnot groups. Prior results in the spirit of this problem include Lang–Plaut’s embedding result for doubling spaces admitting bicombings and geodesic extension properties.
• Bi-Lipschitz embeddings: Hilbert spaces vs. finite-dimensional Euclidean spaces

Problem 2.4.

Suppose that $X$ is a doubling space which admits a bi-Lipschitz embedding into Hilbert space. Does $X$ necessarily admit a bi-Lipschitz embedding into some finite-dimensional Euclidean space?
This is a well known open problem, asked for instance by Naor and Neiman.
• Snowflake embeddings of the Heisenberg group into low-dimensional Euclidean spaces

Let $d_{cc}$ denote the Carnot-Carathéodory metric.

Problem 2.4.

[Enrico Le Donne] Does there exist $\epsilon<1$ so that the first Heisenberg group ${\mathbb H}^1$ equipped with the metric $d_{cc}^\epsilon$ bi-Lipschitz embeds into ${\mathbb R}^5$?
1. Remark. Instead one could use the Korányi metric $d_0(p,q) = ||p^{-1}*q||_0$, where $||(z,t)||_0 = (|z|^4+t^2)^{1/4}$.

Cite this as: AimPL: Mapping theory in metric spaces, available at http://aimpl.org/mappingmetric.