  The purpose of this note is to give a generalization of the statement that
the anticanonical class of a (smooth) projective toric variety is the sum of
invariant prime divisors, corresponding to the rays in its fan (or facets in
its polytope), to some other classes of varieties with algebraic group actions.
To this end, we suggest an analogue of the notion of anticanonical class (of a
compact complex manifold) for linear families of convex polytopes. This is
inspired by the Serre duality for smooth projective varieties as well as the
Ehrhart-Macdonald reciprocity for rational polytopes. The main examples we have
in mind are: (1) The family of polytopes normal to a given fan (which
corresponds to the case of toric varieties). (2) The family of Gelfand-Zetlin
polytopes (which corresponds to the case of the flag variety). (3) The family
of Newton-Okounkov polytopes for a (smooth) group compactification.
