  Stanley (1986) showed how a finite partially ordered set gives rise to two
polytopes, called the order polytope and chain polytope, which have the same
Ehrhart polynomial despite being quite different combinatorially. We generalize
his result to a wider family of polytopes constructed from a poset P with
integers assigned to some of its elements.
  Through this construction, we explain combinatorially the relationship
between the Gelfand-Tsetlin polytopes (1950) and the Feigin-Fourier-Littelmann
polytopes (2010), which arise in the representation theory of the special
linear Lie algebra. We then use the generalized Gelfand-Tsetlin polytopes of
Berenstein and Zelevinsky (1989) to propose conjectural analogues of the
Feigin-Fourier-Littelmann polytopes corresponding to the symplectic and odd
orthogonal Lie algebras.
