2. Embeddings

Deterministic proofs of biLipschitz 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 biLipschitz 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. 
Curvature conditions and biLipschitz embeddings
Problem 2.3.
[Leonid Kovalev] Let $X$ be a doubling metric space. Can one formulate a curvaturetype condition which, if satisfied by $X$, ensures that $X$ admits a biLipschitz embedding into some finitedimensional Euclidean space? 
BiLipschitz embeddings: Hilbert spaces vs. finitedimensional Euclidean spaces
Problem 2.4.
Suppose that $X$ is a doubling space which admits a biLipschitz embedding into Hilbert space. Does $X$ necessarily admit a biLipschitz embedding into some finitedimensional Euclidean space? 
Snowflake embeddings of the Heisenberg group into lowdimensional Euclidean spaces
Let $d_{cc}$ denote the CarnotCarathéodory 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 $d_{cc}^\epsilon$ biLipschitz embeds into ${\mathbb R}^5$?
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.