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A real tree is a metric space $(M,d)$ satisfying the following two conditions:

(1) for every $x,y\in M$, there is an unique isometry $\phi$ from the closed interval $[0,d(x,y)]$ onto $M$ such that $\phi(0)=x$ and $\phi(d(x,y))=y$; and

(2) any one-to-one continuous mapping $f:[0,1]\rightarrow M$ has the same range as the isometry $\phi$ associated to the points $x=\phi(0)$ and $y=\phi(1)$.

Metric spaces that can be isometrically embedded into a real tree are exactly the ones who satisfy the four point condition, that is, metric spaces $(N,\rho)$ such that $$\rho (a,b)+\rho (c,d)\leq max (\rho (a,c)+\rho (b,d),\rho (a,d)+\rho (b,c)) ,a,b,c,d\in N.$$

My question is: what is the most natural/useful way of defining an n-dimensional version of a real tree? Is there any hope to obtain a nice characterization of subspaces, in the spirit of the four point condition?

I would be please to see an answer to this question even for two dimensions.

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One generalization is the 'real cubings' of Casals-Ruiz and Kazachkov. See . – HJRW Dec 13 '12 at 11:02
up vote 4 down vote accepted

See the paper "Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings" MR1608566 by Kleiner and Leeb for a theory of $\mathbb{R}$-buildings that generalizes the theory of $\mathbb{R}$-trees. They use this theory for classifying, up to quasi-isometry, all symmetric spaces of noncompact type whose deRham factors all have rank $\ge 2$. I do not believe they address any characterization such as the 4 point condition.

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