  In this paper, we give a polytopal estimate of Mirkovi\'c-Vilonen polytopes
lying in a Demazure crystal in terms of Minkowski sums of extremal
Mirkovi\'c-Vilonen polytopes. As an immediate consequence of this result, we
provide a necessary (but not sufficient) polytopal condition for a
Mirkovi\'c-Vilonen polytope to lie in a Demazure crystal.
