Definition:Lipschitz Equivalence/Metric Spaces

Definition
Let $M = \left({A, d}\right)$ and $M' = \left({A', d\,'}\right)$ be metric spaces.

Let $f: M \to M'$ be a mapping such that $\exists h, k \in \R: h > 0, k > 0$ such that:
 * $\forall x, y \in A: h d\,' \left({f \left({x}\right), f \left({y}\right)}\right) \le d \left({x, y}\right) \le k d\,' \left({f \left({x}\right), f \left({y}\right)}\right)$

Then $f$ is a Lipschitz equivalence, and $M$ and $M'$ 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