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In section 4.1.3 of Kleiner and Leeb's paper on symmetric spaces and euclidean buildings, there's a result about pairs of rays from the same point initially spanning a flat triangle (or being degenerate).

One way to look view this result is that the following function of distances at $t$ between the rays is initially linear $$d(\alpha(t),\beta(t))\text{ where }\alpha,\beta\text{ are rays with }\alpha(0)=\beta(0)$$ Furthermore, that initial linear part of this function must have a slope according to one of the finitely many angles in what they'd call $D(\theta(\alpha),\theta(\beta))$.

My question:

Is $d(\alpha(t),\beta(t))$ piecewise linear for all $t\in[0,\infty)$?

If so, do the slopes involved correspond to the elements of $D(\theta(\alpha),\theta(\beta))$?

Another way of asking this:

Their argument shows that $\alpha,\beta$ initially span a flat triangle, for example $\alpha(0), \alpha(t_0), \beta(t_0)$.

Can you pick such a $t_0$ and then keep showing that there is $t_1 > t_0$ where you get a flat quadrilateral $\alpha(t_0), \alpha(t_1), \beta(t_1), \beta(t_0)$?

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Pick any time $t_0$ where $\alpha$ and $\beta$ are, locally around $t_0$, each in a fixed chamber : $\alpha(t_0) \in C$ and $\beta(t_0) \in C'$. Then if you consider an apartment $A$ containing $C$ and $C'$, for $t$ close to $t_0$, $d(\alpha(t),\beta(t))$ is just the Euclidean distance between the two rays inside $A$, so in particular it is an affine function.

Furthermore, these small rays inside $A$ also have $\theta(\alpha)$ and $\theta(\beta)$ as asymptotic directions, hence the slope of the distance function also lies in $D(\theta(\alpha),\theta(\beta))$.

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