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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 $G$.

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    $\begingroup$ 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. $\endgroup$
    – Noam D. Elkies
    Commented Sep 30, 2016 at 4:24
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    $\begingroup$ See also: en.wikipedia.org/wiki/Distance_geometry_problem $\endgroup$
    – Noam D. Elkies
    Commented Sep 30, 2016 at 4:25

1 Answer 1

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

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