Polytopal estimate of Mirković–Vilonen polytopes lying in a Demazure crystal

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