Advertisement

On Lipschitz Mappings Onto a Square

Chapter

Abstract

The following problem was posed by Laczkovich [5]: Let \(E \subseteq \mathbb{R}{\mathbb{R}}^{d}\) (d ≥ 2) be a set with positive Lebesgue measure λ d (E) > 0. Does there exist a Lipschitz mapping \(f : {\mathbb{R}}^{d} \rightarrow Q = {[0,1]}^{d}\), such that f(E) = Q? Preiss [6] answered this question affirmatively for d = 2:

Notes

Acknowledgements

I would like to thank David Preiss for explaining me his proof, useful discussions and for his hospitality during my visit.

References

  1. 1.
    G. Alberti, M. Csörnyei, D. Preiss: Structure of null sets in the plane and applications, in European Congress of Mathematics: Stockholm, June 27–July 2, 2004 (A. Laptev ed.), Europ. Math. Soc., Zurich 2005, pages 3–22.Google Scholar
  2. 2.
    G. Alberti, M. Csörnyei, D. Preiss: Differentiability of Lipschitz functions, structure of null sets, and other problems, in Proc. ICM 2010, vol. III, World Scientific, Hackensack, NJ, pages 1379–1394.Google Scholar
  3. 3.
    N. Alon, J. Spencer, P. Erdös: The probabilistic method. Cambridge Univ. Press 1992.MATHGoogle Scholar
  4. 4.
    P. Erdös, G. Szekerés: A combinatorial problem in geometry. Compositio Math. 2(1935) 463–470.MathSciNetGoogle Scholar
  5. 5.
    M. Laczkovich: Paradoxical decompositions using Lipschitz functions, Real Analysis Exchange 17(1991–92), 439–443.Google Scholar
  6. 6.
    D. Preiss, manuscript, 1992.Google Scholar
  7. 7.
    T. Szabó, G. Tardos: A multidimensional generalization of the Erdős–Szekeres lemma on monotone subsequences, Combinatorics, Probability and Computing 10(2001) 557–565.Google Scholar
  8. 8.
    N. X. Uy, Removable sets of analytic functions satisfying a Lipschitz condition, Ark. Mat. 17(1979), 19–27.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    J. H. Wells, L. R. Williams: Embeddings and extensions in analysis, Springer-Verlag 1975.Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Institute of Theoretical Computer Science (ITI)Charles UniversityPragueCzech Republic

Personalised recommendations