Definition:Lipschitz Equivalence/Metric Spaces

Definition
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.

Let $f: M_1 \to M_2$ be a mapping such that $\exists h, k \in \R_{>0}$ such that:
 * $\forall x, y \in A: h d_2 \left({f \left({x}\right), f \left({y}\right)}\right) \le d_1 \left({x, y}\right) \le k d_2 \left({f \left({x}\right), f \left({y}\right)}\right)$

Then $f$ is a Lipschitz equivalence, and $M_1$ and $M_2$ 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 Continuity