  The main object in this paper is a certain rational convex polytope whose
lattice points give a polyhedral realization of a highest weight crystal basis.
This is also identical to a Newton-Okounkov body of a flag variety, and it
gives a toric degeneration. In this paper, we prove that a specific class of
this polytope is given by Kiritchenko's Demazure operators on polytopes. This
implies that polytopes in this class are all lattice polytopes. As an
application, we give a sufficient condition for the corresponding toric variety
to be Gorenstein Fano.
