  In a scenario where two parties share, act on and exchange some physical
resource, the assumption that the parties' actions are ordered according to a
definite causal structure yields constraints on the possible correlations that
can be established. We show that the set of correlations that are compatible
with a definite causal order forms a polytope, whose facets define causal
inequalities. We fully characterize this causal polytope in the simplest case
of bipartite correlations with binary inputs and outputs. We find two families
of nonequivalent causal inequalities; both can be violated in the recently
introduced framework of process matrices, which extends the standard quantum
formalism by relaxing the implicit assumption of a fixed causal structure. Our
work paves the way to a more systematic investigation of causal inequalities in
a theory-independent way, and of their violation within the framework of
process matrices.
