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(Split off from Does every CAT(0) space embed in a measurable integral of $\mathbb{R}$-trees? )

Fix an integer $k \ge 2$, and let $MC0_k \subset \mathbb{R}^{\binom{k}{2}}$ be the set of possible squared-distances between $k$ points (not necessarily distinct) in any CAT(0) space. This is clearly closed under scaling, and the fact that a product of CAT(0) spaces is a CAT(0) space implies that $MC0_k$ is a convex set. What are its extreme rays?

(Recall that an extreme point in a convex set $C$ is a point that is not in the interior of any line segment in $C$. In the context of convex cones, as here, an extreme ray consists of points that aren't in the interior of a line segment that is not contained in a ray through the origin.)

One guess is that the extreme rays in $MC0_k$ are trees with $k$ marked points, where there is at least one marked point at each vertex of valence $>3$. For $k=4$, this includes embeddings in $\mathbb{R}$ as well as tripods, with the central vertex marked.

Note that this is related to the (known-difficult) question of characterizing which lengths can appear as distances in a CAT(0) space, i.e., characterizing $MC0_k$ in the notation above.

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    $\begingroup$ In convex cones it's called "extremal ray". Note that you should allow pseudo-metrics (i.e. collapsing points) to have reasonable compactness properties. $\endgroup$ – YCor May 5 '15 at 11:50
  • $\begingroup$ I actually made the title more vague, leaving the details to the question. $\endgroup$ – Dylan Thurston May 5 '15 at 13:16

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