Definition:Lipschitz Equivalence

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.