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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. 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 $X$.

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In some sense, this is probably the most unsatisfactory answer possible, but it's all I know:

In the 2005 paper 1, 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 H. Movahedi-Lankarani and R. Wells, On bi-Lipschitz embeddings Port. Math. (N.S.) 62 (2005), no. 3, 247–268.

Link to MathSciNet, Link to the paper

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