Definition:Lipschitz Equivalence/Metrics

Definition
Let $M_1 = \left({A, d_1}\right)$ and $M_2 = \left({A, d_2}\right)$ be metric spaces on the same underlying set $A$.

Let $\exists h, k \in \R_{>0}$ such that:
 * $\forall x, y \in A: h d_2 \left({x, y}\right) \le d_1 \left({x, y}\right) \le k d_2 \left({x, y}\right)$

Then $d_1$ and $d_2$ are described as Lipschitz equivalent.

Terminology
Despite the close connection with the concept of Lipschitz continuity, this concept is rarely seen in mainstream mathematics, and appears not to have a well-established name.

The name Lipschitz equivalence appears in :
 * There does not appear to be a standard name for this; the name we use is reasonably appropriate ...

Also see

 * Lipschitz Equivalence is Equivalence Relation


 * Definition:Lipschitz Continuity