Tessellations of an affine apartment by affine weight polytopes

Let $\text{A}$ be a finite dimensional vector space and $\mathrm{\Phi }$ be a finite root system in $\text{A}$. To this data is associated an affine poly-simplicial complex. Motivated by a forthcoming construction of connectified higher buildings, we study “affine weight polytopes” associated to these data. We prove that these polytopes tesselate $\text{A}$. We also prove a kind of “mixed” tessellation, involving the affine weight polytopes and the poly-simplical structure on $\text{A}$.