On a question of Kolmogorov

TL;DR

This paper investigates whether compact planar sets can be approximated by polygons via 1-Lipschitz maps with minimal measure loss, providing positive results for sets with certain boundary properties like the Sierpiński carpet.

Contribution

It offers an equivalent formulation of Kolmogorov's question for compact sets and proves positive results for sets with tube-null boundaries, including the Sierpiński carpet.

Findings

Negative answer for general bounded measurable sets

Positive results for sets with tube-null boundary

Sierpiński carpet can be mapped into line segments with small displacement

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.