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enter image description here

An Isometry in the isomorphism of metric spaces, meaning it preserves all properties of a metric space. Is there a generalization of an Isometry which talks about metric spaces that are the same up to a factor by a scalar? For example, if we look at a circle with radius one and a circle with radius two as sub matric spaces of $R^2$, they are 'almost the same'. They have basically the same structure just that one is factored by two. This is a stronger similarity than just saying that their topological space is isomorphic.

Is there a definition that captures this similarity?

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    $\begingroup$ en.wikipedia.org/wiki/Quasi-isometry $\endgroup$ Aug 16, 2020 at 20:10
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    $\begingroup$ In normed vector spaces, you have the group of dilations (homothety+translation). $\endgroup$
    – Bernard
    Aug 16, 2020 at 20:10
  • $\begingroup$ @MoisheKohan I think that is a weaker definition than what I am talking about, but that is the name I would use :) $\endgroup$
    – BinyaminR
    Aug 16, 2020 at 20:12
  • $\begingroup$ @Bernard that does seem like the same idea of what I am talking about, I wonder why it is defined for a normed space and not a metric one $\endgroup$
    – BinyaminR
    Aug 16, 2020 at 20:16
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    $\begingroup$ I see. As I am not a specialist of topology, I don't know whether it has extensive developments, but I guess it is mainly useful in contexts where there's at least an implicit vector structure. $\endgroup$
    – Bernard
    Aug 16, 2020 at 20:46

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apparently it is called a similitude: enter image description here

from: https://www.wikiwand.com/en/Similarity_(geometry)#/Similarity_in_general_metric_spaces

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  • $\begingroup$ although I'm not sure how the limit there is supposed to be defined $\endgroup$
    – BinyaminR
    Aug 16, 2020 at 20:46
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    $\begingroup$ That's the right definition, although that writer chose an odd word for the terminology. I think most people just call it a similarity. $\endgroup$
    – Lee Mosher
    Aug 16, 2020 at 21:46

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