Defining area / n-volume of a finite metric space

Question

Let $(X, d)$ be a finite metric space. I've seen several answers to the question when can $X$ be isometrically embedded into Euclidean space (or, more generally, Riemannian manifold). I'm interested in whether this idea could be expanded further, so that assuming we know which subsets of points should correspond to "edges", "faces", etc., we could define a reasonable notion of area, volume etc. of $X$.

For example, in one of the answers to Preferred embedding of finite metric spaces in riemaniann manifolds of given dimension it is mentioned that every $4$ element metric space can be isometrically embedded into two model planes (sphere, plane, or Lobachevsky plane, or an R-tree), each with constant curvature. (Unfortunately I wasn't able to find the mentioned result — I will be thankful if someone could suggest where should I look for it).

Therefore, if we were to treat constant curvature of a Riemannian manifold as a good condition, we can define area of such a 4-point metric space by specifying which pairs of points correspond to "edges", and which to "diagonals". After doing so, we could use one of the mentioned embeddings and (hopefully unique) edges to enclose a region of a model plane. The area of this region would be the area of $X$.

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Of course, what would count as a "reasonable notion" is completely subjective, but, the mentioned constant curvature feels somewhat promising. Hence, I would like to know whether there are some sources on the general topic.

Answer 1

Perhaps you are interested in something like Maehara - Euclidean embeddings of finite metric spaces?

For example the Shoenberg theorem mentioned: A finite metric space is embeddable in $R^n$ iff its Gramian matrix is semidefinite and has rank at most $n$.

Also the Matoušek Lecture notes on metric embeddings may be of interest for you. Espesially look on theorem 4.2.1.

In general I don't think that such a question can be solved without the clarification.