Summary
Recently Preiss [4] proved that every subset of the plane of a positive Lebesgue measure can be mapped onto a square by a Lipschitz map. In this note we give an alternative proof of this result, based on a well-known combinatorial lemma of Erdős and Szekeres. The validity of an appropriate generalization of this lemma into higher dimensions remains as an open problem.
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References
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© 1997 Springer-Verlag Berlin Heidelberg
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Matoušek, J. (1997). On Lipschitz Mappings onto a Square. In: Graham, R.L., Nešetřil, J. (eds) The Mathematics of Paul Erdös II. Algorithms and Combinatorics, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60406-5_27
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DOI: https://doi.org/10.1007/978-3-642-60406-5_27
Publisher Name: Springer, Berlin, Heidelberg
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