  Multivariate distributions are fundamental to modeling. Discrete copulas can
be used to construct diverse multivariate joint distributions over random
variables from estimated univariate marginals. The space of discrete copulas
admits a representation as a convex polytope which can be exploited in
entropy-copula methods relevant to hydrology and climatology. To allow for an
extensive use of such methods in a wide range of applied fields, it is
important to have a geometric representation of discrete copulas with desirable
stochastic properties. In this paper, we show that the families of ultramodular
discrete copulas and their generalization to convex discrete quasi-copulas
admit representations as polytopes. We draw connections to the prominent
Birkhoff polytope, alternating sign matrix polytope, and their most extensive
generalizations in the discrete geometry literature. In doing so, we generalize
some well-known results on these polytopes from both the statistics literature
and the discrete geometry literature.
