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On Lipschitz Mappings onto a Square

  • Jiří Matoušek
Part of the Algorithms and Combinatorics book series (AC, volume 14)

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

  1. 1.
    N. Alon, J. Spencer, P. Erdös: The probabilistic method. Cambridge Univ. Press 1992.MATHGoogle Scholar
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    P. Erdös, G. Szekeres: A combinatorial problem in geometry. Compositio Math. 2(1935) 463–470.MathSciNetMATHGoogle Scholar
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    M. Laczkovich: Paradoxical decompositions using Lipschitz functions, Real Analysis Exchange 17(1991–92), 439–443.MathSciNetGoogle Scholar
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    D. Preiss, manuscript, 1992.Google Scholar
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    G. Tardos, private communication, April 1993.Google Scholar
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    J. H. Wells, L. R. Williams: Embeddings and extensions in analysis, Springer-Verlag 1975.MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Jiří Matoušek
    • 1
  1. 1.Department of Applied MathematicsCharles UniversityPraha 1Czech Republic

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