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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.

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  • $\begingroup$ Every finite metric space embeds into $\mathbb R^n$. Maybe try a notion of combined curvature and dimension? $\endgroup$
    – Will Sawin
    Sep 2, 2015 at 15:19
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    $\begingroup$ @Will, definitely not every, if you mean Eucledian metric. $\endgroup$
    – Fedor Petrov
    Sep 2, 2015 at 15:21
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    $\begingroup$ Yes, Fedor is right. For example, take $M=\{x,y,z,t\}$ with $d(x,y)=d(y,z)=d(z,t)=d(t,x)=1$ and $d(x,z)=d(y,t)=2$. This cannot embeds in any euclidean $\mathbb R^n$. $\endgroup$
    – Joël
    Sep 2, 2015 at 15:30
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    $\begingroup$ @Joël Interesting. However, if you take the standard curvature-one $\mathbb{S}^2$ with its standard embedding $\mathbb{S}^2 \subseteq \mathbb{R}^3$, then the points $x'=(1,0,0), y'=(0,1,0), z'=(-1,0,0), t'=(0,-1,0)$ on the equator have the right relative pairwise geodesic distances. So if we scale to a sphere curvature of $\frac{\pi}{2}$, it works. So can that $M$ be said to possess curvature $\frac{\pi}{2}$ in some sense? $\endgroup$
    – Jeppe Stig Nielsen
    Sep 2, 2015 at 19:41
  • $\begingroup$ Possibly related: mathoverflow.net/questions/99505/… $\endgroup$
    – Timothy Chow
    Sep 3, 2015 at 3:24

3 Answers 3

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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. 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 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.

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  • $\begingroup$ Anton, that's exactly what I wanted, thank you very much. $\endgroup$
    – Joël
    Sep 3, 2015 at 15:49
  • $\begingroup$ The link to your book does not work. $\endgroup$
    – Piotr Hajlasz
    Apr 25, 2018 at 16:50
  • $\begingroup$ @PiotrHajlasz fixed $\endgroup$
    – Anton Petrunin
    Apr 29, 2018 at 21:00
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If something more analogous to Ricci curvature than to sectional curvature would interest you, then there has been some work done on the Ricci curvature of discrete spaces. I don't know much about it, but you can find some articles on Yann Olivier's website, and Jürgen Jost and Christian Leonard gave a minicourse on it at the IHS last spring. These might be some starting points for you.

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These notes by John Lott (covering some joint work with Villani) do it for length spaces, which finite metric spaces never are, but if you join the points by edges whose lengths are the distances (so topologically, you have a complete graph), then all is well, and you can use the machinery (which may or may not do what you want).

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  • $\begingroup$ Thank you very much for your answer. I have read these notes and they are very interesting. However, I wonder if the processus of joining any two points with an edge of length their distance is not too destructive of the properties of my space I am interested in. For example I'd like a metric space which is isometric to a subset of a metric tree to have curvature $-\infty$, or at least very low curvature. I am not sure it is still the case if I transform it in a complete graph the way you suggest. I need to think more about it. $\endgroup$
    – Joël
    Sep 2, 2015 at 15:56
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    $\begingroup$ Also, I am not sure there is really new information for me in these notes as compared to all the older literature on $Cat(k)$-spaces. After all, if i understand correctly, the main difference is that they work with length space while Gromov et al work with geidesic space, but for a complete locally compact space, that's the same thing. $\endgroup$
    – Joël
    Sep 2, 2015 at 16:04

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