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On a question of Kolmogorov

Attila Gáspár

TL;DR

This work investigates Kolmogorov's question on whether every bounded planar set can be mapped by a 1-Lipschitz map to a polygon with arbitrarily small measure loss, focusing on compact sets and introducing measure Kolmogorov and distance Kolmogorov as two related notions. It establishes an equivalence between these notions for all $d\ge 2$ by developing the outer measure $\gamma$ built from generalized strips and showing that boundary $\gamma$-nullity implies both properties; a central technical tool is the proximal map framework for polyhedral convex functions. The authors then apply this framework to planar sets with tube-null boundary, proving positive results and in particular mapping the Sierpiński carpet into a finite union of line segments with arbitrarily small displacement. They further show that natural geometric boundaries, such as convex and regular $\mathcal{C}^2$ hypersurfaces, are $\gamma$-null, yielding broad classes of sets that are distance and measure Kolmogorov, and pose several open questions about the limits and invariances of $\gamma$-nullity with potential implications for Kolmogorov-type questions in higher dimensions.

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 Sierpiński 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 Máthé.

On a question of Kolmogorov

TL;DR

This work investigates Kolmogorov's question on whether every bounded planar set can be mapped by a 1-Lipschitz map to a polygon with arbitrarily small measure loss, focusing on compact sets and introducing measure Kolmogorov and distance Kolmogorov as two related notions. It establishes an equivalence between these notions for all by developing the outer measure built from generalized strips and showing that boundary -nullity implies both properties; a central technical tool is the proximal map framework for polyhedral convex functions. The authors then apply this framework to planar sets with tube-null boundary, proving positive results and in particular mapping the Sierpiński carpet into a finite union of line segments with arbitrarily small displacement. They further show that natural geometric boundaries, such as convex and regular hypersurfaces, are -null, yielding broad classes of sets that are distance and measure Kolmogorov, and pose several open questions about the limits and invariances of -nullity with potential implications for Kolmogorov-type questions in higher dimensions.

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 Sierpiński 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 Máthé.
Paper Structure (5 sections, 28 theorems, 34 equations, 2 figures)

This paper contains 5 sections, 28 theorems, 34 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of the equivalence
  3. Generalized strips and the proof of \ref{['gstrip_main']}
  4. Examples of γ-null sets
  5. Open questions

Key Result

Theorem 1.6

Suppose that every compact set in $\mathbb R^d$ ($d \ge 2$) is measure Kolmogorov. Then every compact set in $\mathbb R^d$ is distance Kolmogorov.

Figures (2)

  • Figure 1: The graph of implications between the nullity properties for $d\ge 2$. A crossed arrow indicates that the implication does not hold in general (however, strip-null and tube-null coincide when $d=2$). We do not know whether the implications marked with question marks hold (see \ref{['q_null']}).
  • Figure 2: The generalized strip $S$ associated to $f(x, y) = \max(y, -2y, x - y - 4, -2x + y - 4)$. Outside $S$, $\mathop{\mathrm{prox}}\nolimits_f$ is a translation on each connected component. On the rectangular parts of $S$, $\mathop{\mathrm{prox}}\nolimits_f$ is a projection onto a side combined with a translation. The darker triangles are both mapped to a single point.

Theorems & Definitions (55)

  • Definition 1.2
  • Definition 1.3
  • Definition 1.5
  • Theorem 1.6: Balka, Elekes, Máthé
  • Theorem 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Corollary 1.10
  • Corollary 1.11
  • Corollary 1.12
  • ...and 45 more