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(q1, q2)-quasimetrics bi-Lipschitz equivalent to 1-quasimetrics

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.

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Correspondence to A. V. Greshnov.

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Original Russian Text © A.V. Greshnov, 2017, published in Matematicheskie Trudy, 2017, Vol. 20, No. 1, pp. 81–96.

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Greshnov, A.V. (q1, q2)-quasimetrics bi-Lipschitz equivalent to 1-quasimetrics. Sib. Adv. Math. 27, 253–262 (2017). https://doi.org/10.3103/S1055134417040034

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Keywords

  • distance function
  • (q 1, q 2)-quasimetric
  • generalized triangle inequality
  • extreme point
  • chain approximation
  • Carnot–Carathéodory space