Definition:Lipschitz Equivalence
Definition
Metric Spaces
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $f: M_1 \to M_2$ be a mapping such that $\exists h, k \in \R_{>0}$ such that:
- $\forall x, y \in A: h \map {d_2} {\map f x, \map f y} \le \map {d_1} {x, y} \le k \map {d_2} {\map f x, \map f y}$
Then $f$ is a Lipschitz equivalence, and $M_1$ and $M_2$ are described as Lipschitz equivalent.
Metrics
Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$.
Let $\exists h, k \in \R_{>0}$ such that:
- $\forall x, y \in A: h \map {d_2} {x, y} \le \map {d_1} {x, y} \le k \map {d_2} {x, y}$
Then $d_1$ and $d_2$ 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 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces:
- There does not appear to be a standard name for this; the name we use is reasonably appropriate ...
Sometimes the name strong equivalence (and strongly equivalent metrics) is used.
Also see
- Results about Lipschitz Equivalence can be found here.
Source of Name
This entry was named for Rudolf Otto Sigismund Lipschitz.