Group Norm Metric

A Group Norm Metric is a distance metric on a group ([math]\displaystyle{ G, +, 0 }[/math]), defined by [math]\displaystyle{ \mid\mid x + (-y)\mid\mid= \mid\mid x - y \mid\mid }[/math], where [math]\displaystyle{ \mid\mid . \mid\mid }[/math] is a group norm on G, i.e., a function [math]\displaystyle{ \mid\mid . \mid\mid: G \rightarrow \mathbb{R} }[/math] such that, for all [math]\displaystyle{ x, y \in G }[/math] we have the following properties ...

• See: Order Norm Metric.

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References

2006

• (Deza & Deza, 2006) ⇒ Michel-Marie Deza, and Elena Deza. (2006). “Dictionary of Distances." Elsevier Science. ISBN:9780080465548
  • http://books.google.ca/books?id=I-PQH8gcOjUC&q=group+norm+metric
  • QUOTE: A group norm metrics is a metric on a group ([math]\displaystyle{ G, +, 0 }[/math]), defined by [math]\displaystyle{ \mid\mid x + (-y)\mid\mid= \mid\mid x - y \mid\mid }[/math], where [math]\displaystyle{ \mid\mid . \mid\mid }[/math] is a group norm on G, i.e., a function [math]\displaystyle{ \mid\mid . \mid\mid: G \rightarrow \mathbb{R} }[/math] such that, for all [math]\displaystyle{ x, y \in G }[/math] we have the following properties …

    Any group norm metric d is right-invariant, i.e., …

    … The order norm metric is a group norm metric on G, defined by [math]\displaystyle{ \mid\mid x \cdot y^{-1}\mid\mid_{ord'} }[/math].

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