**Question 1.** Does every CAT(0) space embed isometrically inside an integral of $\mathbb{R}$-trees?


Here an *integral* of $\mathbb{R}$ trees means the set of functions from a measure space $\mathcal{F}$ to a measurable field of based $\mathbb{R}$-trees over $\mathcal{F}$, so that the squared distance from the basepoint is integrable, as described in Proposition 44 and Remark 45 of <https://dx.doi.org/10.1090/S0894-0347-06-00525-X> .


(Originally the question referred to $\ell^2$ products of $\mathbb{R}$-trees, but Nicolas Monod below gave an argument that these do not suffice, and suggested the generalization.)


Really I'm most interested in the finite version of this question (where we also don't need to talk about $\mathbb{R}$-trees):


**Question 2.** Does every $k$-element subset of a CAT(0) space embed isometrically in a (finite) product of metric trees?


To go from Question 2 to Question 1, one would presumably attempt a limiting approximation argument, although it's a little complicated, since an approximation for $k$ points doesn't necessarily have an obvious relation to the approximation for $k+1$ points.


If you consider the set of possible squared-distances between $k$ points in any CAT(0) space, you get a subset $MC0\_k$ of $\mathbb{R}^{\binom{k}{2}}$. This is clearly closed under scaling, and the fact that a product of CAT(0) spaces is still CAT(0) implies that $MC0\_k$ is a convex subset. On the other hand, you could look at $MT\_k$, the set of possible squared-distances between $k$ points in a tre. Question 2 is then asking whether the convex hull of $MT\_k$ is $MC0\_k$. This reformulation makes it clear that, if the answer to Question 2 is affirmative, you would need at most $\binom{k}{2}$ different trees to realize any $k$-element subset of a CAT(0) space.


For $k=4$, the answer to Question 2 appears to be affirmative. Petrunin gives an elegant characterization of which squared-distances between 4 points can be realized in a CAT(0) space: <https://arxiv.org/abs/1411.5329>
Let me sketch the argument.


If the four distances are realized in $\mathbb{E}^3$, we are done. Otherwise, the four distances can (generically) be realized by a space $X\_0$ obtained by gluing together three triangles around a vertex, so that the angle sum around the vertex is greater than $2\pi$. Then consider the space $X\_1$ obtained by deleting one of these triangles. This is again CAT(0), and all distances are the same except between one pair of points. The distances in $X\_0$ can be realized as a convex combination of he distances in $X\_1$ and the distances in another space $X\_3$ where that changed length is shortened until the three triangles embed in $\mathbb{E}^2$.


You can proceed in a similar way, deleting the triangles one at a time, until you reduce $X\_0$ to a convex combination of a tripod and spaces that embed in $\mathcal{E}^2$.


It looks like one could also push this through for $k=5$, although there are many more cases to consider.


*(Moved the question about extreme points here: [What are the extremal CAT(0) metrics?](https://mathoverflow.net/questions/204724/what-are-the-extreme-points-in-the-space-of-cat0-metrics) )*


This is a version of this old question:
[Length inequalities in trees and CAT(0) spaces](https://mathoverflow.net/questions/163706/length-inequalities-in-trees-and-cat0-spaces)
However, I started off asking the earlier question in a slightly confusing dual form, so I wanted to restate it with a more easily-understood question first.


*(Edited to refer to $\mathbb{R}$-trees)*




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Nicolas Monod gave an elegant argument against Question 1 using products. I accepted his answer, but edited the question to refer to integrals instead, which is the natural generalization.


(Or, if you prefer finitary questions, Question 2 is still open, and Nicolas' argument against embedding in a product of trees doesn't really give any evidence against it.)



Does every CAT(0) space embed in a measurable integral of $\mathbb{R}$-trees?
This is not possible; sorry for just posting a sketch but I am not on MO :-(


For instance a generic geodesic triangle in the hyperbolic plane does not embed in a finite product of trees.


Suppose you have a finite product $X$ of uniquely geodesic metric spaces $X\_n$ with the "pythagorean" distance on the product. Then it is uniquely geodesic and any geodesic segment $\sigma\colon[a,b]\to X$ is of the form $\sigma(t)=(\sigma\_n(a\_n t))\_n$ where $a\_n\geq 0$ is a sequence with $\sum a\_n^2 = 1$ and each $\sigma\_n$ is a geodesic in $X\_n$. This characterisation is not entirely trivial, but very well-known (see e.g. Bridson--Haefliger).


Now in the case at hand you would deduce that the distance between two geodesics issuing from a common point would be a piecewise affine function, which it definitely isn't for hyperbolic geometry.


The fact about geodesics extends to infinite sums, and even to integrals (see e.g. Proposition 44 in <https://dx.doi.org/10.1090/S0894-0347-06-00525-X> )


The latter part of the argument above (piecewise affine) does not extend as is to this "integral" setting though.


