  In an earlier paper, the first two authors defined orientations on
hypergraphs. Using this definition we provide an explicit bijection between
acyclic orientations in hypergraphs and faces of hypergraphic polytopes. This
allows us to obtain a geometric interpretation of the coefficients of the
antipode map in a Hopf algebra of hypergraphs. This interpretation differs from
similar ones for a different Hopf structure on hypergraphs provided recently by
Aguiar and Ardila. Furthermore, making use of the tools and definitions
developed here regarding orientations of hypergraphs we provide a
characterization of hypergraphs giving rise to simple hypergraphic polytopes in
terms of acyclic orientations of the hypergraph. In particular, we recover this
fact for the nestohedra and the hyper-permutahedra, and prove it for
generalized Pitman-Stanley polytopes as defined here.
