I want to ask about the progress on **Question 8** from ["Thirty-three yes or no questions about mappings, measures, and metrics" by Juha Heinonen and Stephen Semmes](http://www.ams.org/journals/ecgd/1997-01-01/S1088-4173-97-00012-X/S1088-4173-97-00012-X.pdf). Is it still open? If yes, was some progress achieved? Is it known to be especially hard?


Here is the question:



> 
> **Question 8**. If an Ahlfors regular metric space admits a [David–Semmes] regular map into some
> Euclidean space, then does it admit a bi-Lipschitz map into another, possibly different, Euclidean space?
> 
> 
> 


And definitions:



> 
> A map from one metric space $X$ into another metric space $X'$ is said to be [David–Semmes] **regular** if it is Lipschitz and if there is a constant $C \ge 1$ so that the preimage of each ball of radius $R$ in $X'$ can be covered by at most $C$ balls of radius $R$ in $X$.
> 
> 
> A metric space $X$ is said to be **Ahlfors $s$-regular** for some real number $s > 0$ if
> it has Hausdorff dimension $s$ and if there is a constant $C \ge 1$ such that
> $$C^{−1}R^s ≤ H\_s(B\_R) ≤ CR^s$$
> for each metric ball $B\_R$ of radius $R < diam X$. Here $H\_s$ denotes the $s$-dimensional
> Hausdorff measure in $X$.
> 
> 
> 



Suppose that a metric space allows David–Semmes regular map to some Euclidean space. Does it allow bi-Lipschitz embedding into some Euclidean space?
In some sense, this is probably the most unsatisfactory answer possible, but it's all I know:


In the 2005 paper [1](https://mathscinet.ams.org/mathscinet/search/publdoc.html?pg1=INDI&s1=310836&sort=Newest&vfpref=html&r=4&mx-pid=2171672), Movahedi-Lankarani and Wells write the following in VII (p. 16):



> 
> Tomi Laakso (6) has informed us that his non-embeddability theorem [9] can be generalized in such a way that the non-embeddability theorem of Cheeger [4] no longer implies it. In a forthcoming paper, he gives a sufficient condition for a metric space to admit no bi-Lipschitz embedding into uniformly convex Banach spaces. Furthermore, he shows that there exists a metric space which does not admit a bi-Lipschitz embedding into any uniformly convex Banach space even though it admits a David-Semmes regular mapping onto $\mathbb{R}^2$.
> 
> 
> 


Reference [9] is to the paper of Laakso in which he constructs Ahlfors $Q$-regular spaces of arbitrary $Q>1$ admitting Poincare inequalities and not bi-Lipschitz embedding into any uniformly convex Banach space (which was published in GAFA). Unfortunately, the footnote (6) is just "private communication". As far as I know, the "forthcoming paper" of Laakso mentioned in this quote never came out.


I for one would be happy to see an example like this, but I have not seen one anywhere else. Of course, someone might know better.


[1](https://mathscinet.ams.org/mathscinet/search/publdoc.html?pg1=INDI&s1=310836&sort=Newest&vfpref=html&r=4&mx-pid=2171672) H. Movahedi-Lankarani and R. Wells, *On bi-Lipschitz embeddings* Port. Math. (N.S.) 62 (2005), no. 3, 247–268.


[Link to MathSciNet](https://mathscinet.ams.org/mathscinet/search/publdoc.html?pg1=INDI&s1=310836&sort=Newest&vfpref=html&r=4&mx-pid=2171672), [Link to the paper](https://www.emis.de/journals/PM/62f3/pm62f301.pdf)


