Mirković–Vilonen polytopes lying in a Demazure crystal and an opposite Demazure crystal

We give a necessary and sufficient condition for an MV polytope P in a highest weight crystal to lie in a fixed (but arbitrary) Demazure crystal (resp., opposite Demazure crystal), in terms of the lengths of edges along a path through the 1-skeleton of P corresponding to a reduced word for the longest element of the Weyl group W. Also, we give an explicit description as a pseudo-Weyl polytope of the extremal MV polytopes in a highest weight crystal. Finally, by combining the results above, we obtain a polytopal condition for an MV polytope P to lie in a fixed (but arbitrary) opposite Demazure crystal.