On Lipschitz Mappings Onto a Square



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:



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


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© 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

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