Definition:Lipschitz Equivalence

Metric Spaces
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 $$\exists h, k \in \reals: h > 0, k > 0$$ 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.

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

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

Then $$d$$ and $$d'$$ are described as Lipschitz equivalent.

This is clearly an equivalence relation.

If we consider the identity mapping $$f: A \to A: \forall x \in A: f \left({x}\right) = x$$, we can likewise directly consider $$f: \left\{{A, d}\right\} \to \left\{{A, d'}\right\}$$ as a Lipschitz equivalence.