simplex containing the point in its interior. We will sometimes denote the simplex spanned by $\xi$ by $\tau_{\xi}$.

A spherical building is [*thick*]{} if every wall is the bounds at least three half-apartments, equivalently, if every panel is adjacent to (i.e. contained in the boundary of) at least three chambers. One can always pass to a thick spherical building structure by reducing the Weyl group, thereby coarsifying the simplicial structure.

The [*space of directions*]{} $\Si_\xi \B$ at a point $\xi\in \B$ is the space of germs $\oa{\xi\eta}$ of nondegenerate geodesic segments $\xi\eta\subset \B$, equipped with the natural angle metric $\angle_{\xi}$. Two segments $\xi\eta$ and $\xi\eta'$ represent the same direction in $\Si_\xi \B$, $\oa{\xi\eta}=\oa{\xi\eta'}$, iff they initially agree. The space of directions is again a spherical building.

A subset $C\subset \B$ is called ($\pi$-)[*convex*]{} if for any two points $\xi, \eta\in C$ with distance $\angle(\xi,\eta)<\pi$ the (unique) geodesic $\xi\eta$ connecting $\xi$ and $\eta$ in $\B$ is contained in $C$.

Due to the compatibility of charts, i.e. the property of the building atlas that the coordinate changes are induced by isometries in $W$, there is a well-defined [*type*]{} map $$\theta: \B\to\simod .$$ It is 1-Lipschitz and restricts to an isometry on every 