We say that a matrix $M\in\mathbb{R}^{n\times n}$ is a distance matrix on a metric space $(X,d)$, if there exist $x\_1,\cdots,x\_n \in X$ such that $M=[d(x\_i,x\_j)]\_{n\times n}$.


**Question.** For which metric spaces are the properties of distance matrices investigated?



Distance matrices
The first paper to read is "[Metric spaces and positive definite functions](http://datamin.ubbcluj.ro/semi/pdfs/Schoenberg.pdf)" by Schoenberg. (I first learned about this from Igor Rivin many years ago.) It follows from his main result + Corollary 1 that a (say, finite) metric space $(M,d)$ embeds isometrically in a Hilbert space if and only if the "distance matrix" of $(M,d)$ is conditionally negative definite. (Schoenberg first proves it for the matrix of square distances, but then Corollary 1 allows to get rid of squares.) 


One can ask the same question about other spaces, like the separable infinite-dimensional real-hyperbolic space. In this setting, there are 4-point conditions on the matrix of distances (coming from negative curvature); but these conditions are insufficient and the problem is open. On the other hand, if one asks for an isometric embedding in the "universal metric tree" of Dyubina/Erschler and Polterovich, then the necessary and sufficient condition is known: It is equivalent to $0$-hyperbolicity of $(M,d)$, which is a combination of triangle inequalities and some 4-point conditions on the distance matrix. 


On yet another hand, if one relaxes isometric to quasi-isometric embeddings but considers maps to real-hyperbolic spaces, then the problem was solved (for domains which are "reasonable" Gromov-hyperbolic spaces) by a deep theorem of M.Bonk and O.Schramm in the paper "[Embeddings of Gromov hyperbolic spaces](http://link.springer.com/article/10.1007/s000390050009#page-1)". 


Lastly, I proved some time ago a general theorem about metric matrix for geodesic metric spaces $(M,d)$: Every triple of distances (subject to triangle inequalities) is realized by a geodesic triangle in $(M,d)$ if and only if $M$ is not quasi-isometric to point, half-line and the line. See [here](http://www.math.ucdavis.edu/~kapovich/EPR/tri.pdf). This settles the question in the case $n=3$ for a reasonable class of metric spaces $(M,d)$. 


