Strongly dominant weight polytopes are cubes

For any root system of rank $r$, we study the “dominant weight polytope” $P^{\lambda }$ associated with a strongly dominant weight $\lambda$. We prove that $P^{\lambda }$ is combinatorially equivalent to the $r$-dimensional cube. As an application, we give a new proof of the known formulas for Betti numbers of Peterson varieties in classical Lie types.