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I have distance matrix $D$ that was calculated by some distance (non-Euclidean but satisfying distance requirements). Is there a set of points in some Euclidean space such that it generates matrix of Euclidean distances that is equal $D$?

I know that if $G=-HDH/2$ is p.d. where H is the centering matrix then such embedding exists. However, I don't have any information about $G$.

Dmitry Marin
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    Choose one of the points are the origin. Use the "polarization identity" to write find $(x,y)$ for any other points $x,y$ using the known distances $|x|^2$, $|y|^2$, $|x+y|^2$. The embedding problem can be solved iff the resulting Gram matrix is positive-definite. – Noam D. Elkies Sep 30 '16 at 04:24
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    See also: https://en.wikipedia.org/wiki/Distance_geometry_problem – Noam D. Elkies Sep 30 '16 at 04:25

1 Answers1

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No. The magic words are Cayley-Menger, but an explicit answer is given here.

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Igor Rivin
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