copies of ${\mathbb{R}}^n$ so is the *model space* ${\mathbb{A}}$ of a generalized affine building isomorphic to $\Lambda^n$. We define $${\mathbb{A}}({\Phi},\Lambda) = \mathrm{span}_F({\Phi})\otimes_F \Lambda,$$ where $F$ is a sub-field of the reals containing all evaluations of co-roots on roots.

The spherical Weyl group ${\overline{W}}$ associated to ${\Phi}$ acts on ${\mathbb{A}}$. A *hyperplane* $H_\alpha$ in the model space is a fixed point set of a reflection $r_\alpha$ in ${\overline{W}}$ which separates ${\mathbb{A}}$ into two half-spaces, called *half-apartments*. There is as well an affine Weyl group ${{{W}}_T}$ acting on ${\mathbb{A}}$, which is the semi-direct product of ${\overline{W}}$ by some ${\overline{W}}$ invariant translation group $T$ of the model space. In case the translation group $T$ is the entire space ${\mathbb{A}}$ we write ${W}$ instead of ${{{W}}_T}$.

Associated to a basis $B$ of the root system ${\Phi}$ there is a *fundamental Weyl chamber* ${\mathcal{{C}}_{f}}$. The chamber ${\mathcal{{C}}_{f}}$ is a fundamental domain for the action of ${\overline{W}}$ on ${\mathbb{A}}$ and its images under the affine Weyl group are the *Weyl chambers* in ${\mathbb{A}}$. A *Weyl simplex* is a face of a Weyl chamber. The smallest face of dimension 0 is called *basepoint*.

One can endow ${\mathbb{A}}$ with a natural ${W}$-invariant metric taking its values in $\Lambda$ 