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(q 1, q 2)-quasimetrics bi-Lipschitz equivalent to 1-quasimetrics. / Greshnov, A. V.

In: Siberian Advances in Mathematics, Vol. 27, No. 4, 01.10.2017, p. 253-262.

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Greshnov, AV 2017, '(q 1, q 2)-quasimetrics bi-Lipschitz equivalent to 1-quasimetrics', Siberian Advances in Mathematics, vol. 27, no. 4, pp. 253-262. https://doi.org/10.3103/S1055134417040034

APA

Greshnov, A. V. (2017). (q 1, q 2)-quasimetrics bi-Lipschitz equivalent to 1-quasimetrics. Siberian Advances in Mathematics, 27(4), 253-262. https://doi.org/10.3103/S1055134417040034

Vancouver

Greshnov AV. (q 1, q 2)-quasimetrics bi-Lipschitz equivalent to 1-quasimetrics. Siberian Advances in Mathematics. 2017 Oct 1;27(4):253-262. doi: 10.3103/S1055134417040034

Author

Greshnov, A. V. / (q 1, q 2)-quasimetrics bi-Lipschitz equivalent to 1-quasimetrics. In: Siberian Advances in Mathematics. 2017 ; Vol. 27, No. 4. pp. 253-262.

BibTeX

@article{9c5a4b4f38a341878b20363f573191c1,
title = "(q 1, q 2)-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. Schr{\"o}der.",
keywords = "(q, q)-quasimetric, Carnot–Carath{\'e}odory space, chain approximation, distance function, extreme point, generalized triangle inequality",
author = "Greshnov, {A. V.}",
year = "2017",
month = oct,
day = "1",
doi = "10.3103/S1055134417040034",
language = "English",
volume = "27",
pages = "253--262",
journal = "Siberian Advances in Mathematics",
issn = "1055-1344",
publisher = "PLEIADES PUBLISHING INC",
number = "4",

}

RIS

TY - JOUR

T1 - (q 1, q 2)-quasimetrics bi-Lipschitz equivalent to 1-quasimetrics

AU - Greshnov, A. V.

PY - 2017/10/1

Y1 - 2017/10/1

N2 - 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.

AB - 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.

KW - (q, q)-quasimetric

KW - Carnot–Carathéodory space

KW - chain approximation

KW - distance function

KW - extreme point

KW - generalized triangle inequality

UR - http://www.scopus.com/inward/record.url?scp=85036551701&partnerID=8YFLogxK

U2 - 10.3103/S1055134417040034

DO - 10.3103/S1055134417040034

M3 - Article

AN - SCOPUS:85036551701

VL - 27

SP - 253

EP - 262

JO - Siberian Advances in Mathematics

JF - Siberian Advances in Mathematics

SN - 1055-1344

IS - 4

ER -

ID: 9648781