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 -