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

Polytopal Estimate of Mirkovic-Vilonen polytopes lying in a Demazure crystal