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Laczkovich Lipschitz Mapping Problem

The Laczkovich Lipschitz mapping problem asks whether every measurable set A subset= R^d of positive Lebesgue measure admits a Lipschitz surjection onto the unit hypercube [0,1]^d (Balka and Keleti 2024). The Lipschitz constant may depend on A.

Gáspár (2026) solved the problem by proving that there is a global 1-Lipschitz map f:R^d->R^d such that f(A) has nonempty interior. A sufficiently small cube in that image can be projected onto and rescaled to the unit cube. Conversely, a Lipschitz image containing a unit cube has nonempty interior. Thus the global 1-Lipschitz formulation does not assert a uniform size for the cube. Gáspár (2026) attributes proofs of key intermediate results to ChatGPT and states that he rewrote and checked the arguments.

See also

Lipschitz Function, Lipschitz Constant

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References

Balka, R. and Keleti, T. "Lipschitz Images and Dimensions." Adv. Math. 446, 109669, 2024. https://doi.org/10.1016/j.aim.2024.109669.Gáspár, A. "Solution of Laczkovich's Lipschitz Mapping Problem." 28 Oct 2026. https://arxiv.org/abs/2610.38834.

Cite this as:

Weisstein, Eric W. "Laczkovich Lipschitz Mapping Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LaczkovichLipschitzMappingProblem.html

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