I am sorry to ask a very vague question, but:



> 
> What are good ways to define the curvature of a finite metric space?
> 
> 
> 


The best way I can think of is: the curvature of a finite metric space $M$
is the infimum of the real $k$ such that there is a geodesic metric space $X$ which is $Cat(k)$ and $M$ embeds isometrically in $X$. However, I don't know how to compute this curvature form the distance matrice... Is there a way? Moreover
is this notion defined somewhere, and has its properties being studied? 
I would also be open to any other sensible definition, or reference discussing this question. Thanks.


PS: I ask this question for experimental purposes. I have a metric set of data I would like to define and compute the curvature of.



Curvature of a finite metric space
It is not a problem to define, Alexandrov's comparison inequalities make sense for all metric spaces. Check (3+1) and (2+2) point comparison in [our book](http://arxiv.org/abs/1903.08539). However, it is not clear what to do with these spaces (I do not know anything interesting about them unless they have length metric.)


A different definition (defining bigger class of spaces) is given by Nikolaev and Berg, see [this paper](http://arxiv.org/abs/1008.3318) and the references there in.


The question which spaces admit a distance preserving map into CATs and CBBs is open, it is discussed [here in section 7](http://arxiv.org/abs/1012.5636).


