0

If we have $n$ points in some metric space, do there exist coordinates for the $n$ points in an $n-1$ dimensional Euclidean space with exactly the same pairwise distances as in the original space?

This post gives a counterexample. But is there a condition on the finite metric space that is less strong than it being euclidean to ensure a perfect embedding is possible?

Anne Bauval
  • 49,005
user9998990
  • 1
  • Thank you. Would you mind a sketch? I think I see it, progressively add a point, and each addition is possible by the triangle inequality, but I don't know if this guarantees all pairwise distances are preserved.

    It seems like there would be a well known theorem or something for this question.

     – user9998990 Jul 20 '24 at 00:08
  • I think this relates to multidimensional scaling, but I cant find a theorem or reference saying that an exact preservation of distances is possible. – user9998990 Jul 20 '24 at 00:14
  • Look at this image: https://i.sstatic.net/AJCbmGI8.png – Lucenaposition Jul 20 '24 at 00:22
  • That image is a helpful counterexample, thanks. Is there a condition on the finite metric space that is less strong than it being euclidean to ensure a perfect embedding is possible? – user9998990 Jul 20 '24 at 04:39
  • Still a duplicate: https://math.stackexchange.com/questions/3557030/how-many-dimensions-we-need-to-represent-weighted-graph-in-metric-space/3563638#3563638 – Moishe Kohan Jul 20 '24 at 04:54
  • The image was less detailed than the previously linked post. Your post was closed as a duplicate. Now you seem to have a new question but 1) imo you should open a new post for it instead of drastically modifying this one, which makes the closure non-understandable (please roll back to your original question) 2) In that new post, better clarify what you mean by "condition on the finite metric space that is less strong than it being euclidean". – Anne Bauval Jul 20 '24 at 06:33

0 Answers0