Characterization of compact subsets of curves with $\omega$-continuous derivatives

Tom 211 / 2011

Marcin Pilipczuk Fundamenta Mathematicae 211 (2011), 175-195 MSC: Primary 53A04. DOI: 10.4064/fm211-2-4


We give a characterization of compact subsets of finite unions of disjoint finite-length curves in $\mathbb R^n$ with $\omega$-continuous derivative and without self-intersections. Intuitively, our condition can be formulated as follows: there exists a finite set of regular curves covering a compact set $K$ iff every triple of points of $K$ behaves like a triple of points of a regular curve.

This work was inspired by theorems by Jones, Okikiolu, Schul and others that characterize compact subsets of rectifiable or Ahlfors-regular curves. However, their classes of curves are much wider than ours and therefore the condition we obtain and our methods are different.


  • Marcin PilipczukInstitute of Informatics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland

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