(q 1, q 2)-quasimetrics bi-Lipschitz equivalent to 1-quasimetrics

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Abstract

We prove that the conditions of (q1, 1)- and (1, q2)-quasimertricity of a distance function ρ are sufficient for the existence of a quasimetric bi-Lipschitz equivalent to ρ. It follows that the Box-quasimetric defined with the use of basis vector fields of class C1 whose commutators at most sum their degrees is bi-Lipschitz equivalent to some metric. On the other hand, we show that these conditions are not necessary. We prove the existence of (q1, q2)-quasimetrics for which there are no Lipschitz equivalent 1-quasimetrics, which in particular implies another proof of a result by V. Schröder.

Original languageEnglish
Pages (from-to)253-262
Number of pages10
JournalSiberian Advances in Mathematics
Volume27
Issue number4
DOIs
Publication statusPublished - 1 Oct 2017

Keywords

  • (q, q)-quasimetric
  • Carnot–Carathéodory space
  • chain approximation
  • distance function
  • extreme point
  • generalized triangle inequality

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