  Consider a root system $R$ and the corresponding toric variety $V_R$ whose
fan is the Weyl fan and whose lattice of characters is given by the root
lattice for $R$. We prove the vanishing of the higher cohomology groups for
certain line bundles on $V_R$ by proving a purely combinatorial result for root
systems. These results are related to a converse to Mazur's Inequality for
(simply-connected) split reductive groups.
