Geodesic
$(q_1,q_2)$-quasimetrics bi-Lipschitz equivalent to $1$-quasimetrics
Matematičeskie trudy, Tome 20 (2017) no. 1, pp. 81-96

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We prove that the conditions of $(q_1,1)$- and $(1,q_2)$-quasimertricity of a distance function $\rho$ are sufficient for the existence of a quasimetric bi-Lipschitz equivalent to $\rho$. It follows that the Box-quasimetric defined with the use of basis vector fields of class $C^1$ 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 $(q_1,q_2)$-quasimetrics for which there are no Lipschitz equivalent $1$-quasimetrics, which in particular implies another proof of a result by V. Schröder. 
@article{MT_2017_20_1_a4,
     author = {A. V. Greshnov},
     title = {$(q_1,q_2)$-quasimetrics {bi-Lipschitz} equivalent to $1$-quasimetrics},
     journal = {Matemati\v{c}eskie trudy},
     pages = {81--96},
     publisher = {mathdoc},
     volume = {20},
     number = {1},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2017_20_1_a4/}
}
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A. V. Greshnov. $(q_1,q_2)$-quasimetrics bi-Lipschitz equivalent to $1$-quasimetrics. Matematičeskie trudy, Tome 20 (2017) no. 1, pp. 81-96. http://geodesic.mathdoc.fr/item/MT_2017_20_1_a4/