Abstract: Given a crystallographic root system with a fixed simple system, it is associated to a Weyl group $W$ and a polytope $P$ that is the convex hull of the $W$-orbit of a well-chosen dominant weight. The quotient $P/W$ can be identified with a polytope. Polytopes $P$ and $P/W$ are associated to toric varieties $X_P$ and $X_{P/W}$ respectively. It turns out the underlying topological spaces $X_P/W$ and $X_{P/W}$ are homotopy equivalent.