Tamás Keleti & Attila Gáspár: Solution of Laczkovich's Lipschitz mapping problem

Abstract: In 1991, Laczkovich asked whether every measurable set in \(\mathbb{R}^d\) can be mapped onto a cube with a Lipschitz map. The case \(d = 2\) was first proved by Preiss. Later, Matoušek gave a stronger version of this result. However, the case \(d \geq 3\) was open until now. In this talk, we will see that the answer to Laczkovich’s question is positive for every \(d\). The proof uses some tools from convex analysis, most importantly, the proximal map of a convex function. A key step in the proof was done by ChatGPT, however, the overall proof stategy is based on my research on a related problem by Kolmogorov, which asks whether a set can be mapped onto a polygon with a contraction such that the measure loss is arbitrarily small.