Definition:Lipschitz Equivalence/Metric Spaces

Definition
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.

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

Then $f$ is a Lipschitz equivalence, and $M_1$ and $M_2$ are described as Lipschitz equivalent.

Also see

 * Definition:Lipschitz Continuity