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.

Source of Name

This entry was named for Rudolf Lipschitz.

Sources

• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Proposal $2.4.3$, Definition $2.4.8$