Definition:Lipschitz Equivalence

Metric Spaces
Let $$M = \left({A, d}\right)$$ and $$M^{\prime} = \left({A^{\prime}, d^{\prime}}\right)$$ be metric spaces.

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

Then $$f$$ is a Lipschitz equivalence, and $$M$$ and $$M^{\prime}$$ are described as Lipschitz equivalent.

Metrics
Let $$A$$ be a set upon which there are two metrics imposed: $$d$$ and $$d^{\prime}$$.

Let $$\exists h, k \in \R: h > 0, k > 0$$ such that $$\forall x, y \in A: h d^{\prime}\left({x, y}\right) \le d \left({x, y}\right) \le k d^{\prime}\left({x, y}\right)$$.

Then $$d$$ and $$d^{\prime}$$ are described as Lipschitz equivalent.

This is clearly an equivalence relation.