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 language | English |
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Pages (from-to) | 253-262 |
Number of pages | 10 |
Journal | Siberian Advances in Mathematics |
Volume | 27 |
Issue number | 4 |
DOIs | |
Publication status | Published - 1 Oct 2017 |
Keywords
- (q, q)-quasimetric
- Carnot–Carathéodory space
- chain approximation
- distance function
- extreme point
- generalized triangle inequality