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Question. If $A\subset \mathbb{R}^n$ is any set of positive Lebesgue $n$-measure, does there exists a Lipschitz map $f:A\to\mathbb{R}^n$ such that $f(A)$ is a ball with the same measure?

In dimension $n=1$, such a map is easily found by defining $f(x) = \mathcal{L}^1(A\cap [0,x])$. In higher dimensions the problem becomes much more difficult. There is a short discussion on the topic in Ch. 7.10 of Mattila's 1995 book Geometry of Sets and Measures in Euclidean Spaces, where it is said that

"Recently Preiss [6] proved that if $A$ is an $\mathcal{L}^2$ measurable subset of $\mathbb{R}^2$ with $\mathcal{L}^2(A) > 0$, there is a Lipschitz map $f:A\to \mathbb{R}^2$ such that $fA$ is a disc. In $\mathbb{R}^n$ for $n\geq 3$ the problem is unsolved."

The reference given is

[6] D. Preiss. Lipschitzian images of planar sets of positive measure, preprint

However I was unable to find the mentioned paper of Preiss anywhere. At least no paper with such a title seems to exist on MathSciNet, arXiv or Preiss's list of publications. Given that Mattila referred to the result as "recent" in 1995, it seems unlikely that the paper would still be a preprint, so a few possibilities come to mind:

  • The title changed before publication
  • An error in the proof was found and the paper was never published
  • The result was included in a different paper

Any information about what has happened to the paper of Preiss, or any other references or information about the current state of the problem would be much appreciated.

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  • $\begingroup$ Can't you apply the construction of $n=1$ to the one coordinate, iterate and get donw to a cube? $\endgroup$ – user1688 Sep 6 '18 at 11:53
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The problem is also mentioned in

Alberti, Giovanni and Csörnyei, Marianna and Preiss, David: Structure of null sets in the plane and applications. European Congress of Mathematics, 3–22, Eur. Math. Soc., Zürich, 2005. MR2185733. PDF

They note that D. Preiss has proved it (in a forthcoming paper) but note that it has also been proved by

Matoušek, Jiří: On Lipschitz mappings onto a square. The mathematics of Paul Erdős, II, 303–309, Algorithms Combin., 14, Springer, Berlin, 1997. MR1425223 ps.gz

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