Marginal polytopes are important geometric objects that arise in statistics as the
polytopes underlying hierarchical log-linear models. These polytopes can be used to
answer geometric questions about these models, such as determining the existence of
maximum likelihood estimates or the normality of the associated semigroup. Cut
polytopes of graphs have been useful in analyzing binary marginal polytopes in the
case where the simplicial complex underlying the hierarchical model is a graph. We
introduce a generalized cut polytope that is isomorphic to the binary marginal
polytope of an arbitrary simplicial. This polytope is full dimensional in its ambient
space and has a natural switching operation among its facets that can be
used to deduce symmetries between the facets of the correlation and binary
marginal polytopes. We use this switching operation along with Bernstein and
Sullivant’s characterization of unimodular simplicial complexes to find a complete
-representation
for many unimodular simplicial complexes.