Let $X$ be a CAT(0) space, $x \in X$ and $\xi$ be a point in the boundary at infinity of $X$. Denote by $\rho(t)$ the geodesic ray (parameterized by arc length) starting from $x$ and asymptotic to $\xi$. Let $y$ be a point close to $x$ and let $\overline{xy}$ denote the geodesic segment connecting $x$ and $y$. I would like to estimate the angle (at $x$) between $\overline{xy}$ and $\rho$, in terms of the Busemann function
$$b\_{\xi}(y) = \lim\_{t \to \infty} d(y, \rho(t)) -t.
$$
Precisely, I would like to say that if $b\_{\xi}(y)>0$, then the angle is obtuse.


Can this be true? I’m interested in the case where $X$ is a Euclidean building, but I was hoping it is a known fact (either true or false) about CAT(0) spaces.



Angles and Busemann function in CAT(0)
The answer is "no" even in the hyperbolic plane $\mathbb{H}^2$.


Consider the horocycle about $\xi$ through $x$: this is the set of points such that $b\_{\xi}(z)=0$. Let $\gamma$ be the geodesic through $x$ tangent to the horocycle, and take $y$ to be any point on $\gamma$ (except $x$ itself). Then the angle between $\overline{xy}\subset\gamma$ and $\rho$ is $\pi/2$, but $b\_{\xi}(y)>0$ since $y$ is outside the horodisc enclosed by the horocycle.


However, similar geometric arguments show that a slightly weaker statement is true in $\mathbb{H}^2$, which may be covered by your assumption that $y$ is 'close enough' to $x$, as follows. For any $\theta<\pi/2$ there is an $\epsilon>0$ such that, if $d(x,y)<\epsilon$ and the angle between $\rho$ and $\overline{xy}=\theta$, then $b\_{\xi}(y)<0$.


