  We study a family of posets and the associated chain and order polytopes. We
identify the order polytope as a maximal Kogan face in a Gelfand-Tsetlin
polytope of a multiple of a fundamental weight. We show that the character of
such a Kogan face equals to the character of a Demazure module which occurs in
the irreducible representation of $\mathfrak{sl}_n$ having highest weight
multiple of fundamental weight and for any such Demazure module there exists a
corresponding poset and associated maximal Kogan face. We prove that the chain
polytope parametrizes a monomial basis of the associated PBW-graded Demazure
module and further, that the Demazure module is a favourable module, e.g.
interesting geometric properties are governed by combinatorics of convex
polytopes. Thus, we obtain for any minuscule Schubert variety a flat
degeneration into a toric projective variety which is projectively normal and
arithmetically Cohen-Macaulay. We provide a necessary and sufficient condition
on the Weyl group element such that the toric variety associated to the chain
polytope and the toric variety associated to the order polytope are isomorphic.
