example, $(11)$ describes the root polytope, the hexagon with 2 points in the middle in Figure \[A2polytopes\], and each point corresponds to one of the 8 states of the adjoint representation, so $(11)$ can refer also to the whole adjoint representation. The fundamental simplex of the lattice $A_{n}$ is a convex polytope with $n+1$ vertices given by $\omega_{1},...,\omega_{n}$ and the origin $(0)$. The Voronoi polytope $V(0)$ centered at the origin of the lattice $\Lambda$ is the set of points $V(0)=\left\{ x\in\mathbb{R}^{n},\forall p,(x,p)\leq\frac{1}{2}(p,p)\right\} $. Let $v\in\mathbb{R}^{n}$ be a vertex of an arbitrary Voronoi polytope $V(p)$. The convex hull of the lattice points closest to $v$ is called the Delone polytope containing $v$. Vertices of the Voronoi cell $V(0)$ of the root lattices of the $A_{n}$ series consist of the vertices of the Delone cells centered at the origin.

Let us show the analogue of these polytopes in lower dimension for $A_1=SU(2)$. First write the eigenvalues in integer form, $p=2j$ and then:

-   $p=1$ is the highest weight of the fundamental representation with one reflection that sends it to the state $p=-1$. The edge from the $p=-1$ to $p=1$ is the Voronoi polytope, written $(1)$ in the notation above.

-   With $p=2$, 