Definition:Lipschitz Equivalence/Metrics/Definition 1
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Definition
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
Source of Name
This entry was named for Rudolf Otto Sigismund Lipschitz.
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.4$: Equivalent metrics: Proposal $2.4.3$