On a question of Kolmogorov

Abstract

Kolmogorov asked the following question: can every bounded measurable set in the plane be mapped onto a polygon by a 1-Lipschitz map with arbitrarily small measure loss? The answer is negative in general, however, the case of compact sets is still open. We present an equivalent form of the question for compact sets. Furthermore, we give a positive answer to Kolmogorov's question for specific classes of sets, most importantly, for planar sets with tube-null boundary. In particular, we show that the Sierpi\'nski carpet can be mapped into the union of finitely many line segments by a 1-Lipschitz map with arbitrarily small displacements, answering a question of Balka, Elekes and M\'ath\'e.