This result addresses Kolmogorov's 1932 question of whether a bounded measurable planar set can be sent onto a polygon by a 1-Lipschitz (nonexpansive) map while losing arbitrarily little Lebesgue measure. Although the answer is negative for general bounded sets and open for compact sets, a positive answer holds when the set's boundary is tube-null: such planar sets can be 1-Lipschitz mapped onto polygons with arbitrarily small measure loss. The framework also yields an equivalent reformulation of the compact case and a higher-dimensional polyhedron analogue, and it implies that the Sierpinski carpet can be carried into a finite union of line segments by a 1-Lipschitz map with arbitrarily small displacements.
This paper advances Kolmogorov's 1932 question in metric geometry: can every bounded measurable planar set be mapped onto a polygon by a 1-Lipschitz map while losing arbitrarily little Lebesgue measu…