Definition:Lipschitz Equivalence/Metrics

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Definition

Definition 1

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.


Definition 2

Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$.

Let $\exists K_1, K_2 \in \R_{>0}$ such that:

$(1): \quad \forall x, y \in A: \map {d_2} {x, y} \le K_1 \map {d_1} {x, y}$
$(2): \quad \forall x, y \in A: \map {d_1} {x, y} \le K_2 \map {d_2} {x, y}$


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 ...


Also see


Source of Name

This entry was named for Rudolf Lipschitz.