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 http://dx.doi.org/10.1090/S0894-0347-06-00525-X .

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: http://front.math.ucdavis.edu/1411.5329 Let me sketch the argument.

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 .

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.

(Moved the question about extreme points here: What are the extremal CAT(0) metrics? )

This is a version of this old question: Length inequalities in trees and CAT(0) 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.

(Moved the question about extreme points here: What are the extremal CAT(0) metrics? )

This is a version of this old question: Length inequalities in trees and CAT(0) 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.

Question 1. Does every CAT(0) space embed isometrically inside an (infinite) product (or integral, see below) of $\mathbb{R}$-trees?

Here the product of two metric spaces has the $\ell^2$ square-root-of-sum-of-squares metric, as in Euclidean space. Note that $\mathbb{R}$ is a tree, so any convex subset of Euclidean space embeds in a product of trees, but of course most CAT(0) spaces are do not embed in Euclidean space.

Nicolas Monod gave an elegant argument against Question 1. However, in the same answer, he also suggests a generalization where you look at embeddings in a measurable integral of trees. So I accepted his answer, but I'm also interested in the generalization.

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 http://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.)

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.