  All correlation measures, classical and quantum, must be monotonic under
local operations. In this paper, we characterize monotonic formulas that are
linear combinations of the von Neumann entropies associated with the quantum
state of a physical system that has n parts. We show that these formulas form a
polyhedral convex cone, which we call the monotonicity cone, and enumerate its
facets. We illustrate its structure and prove that it is equivalent to the cone
of monotonic formulas implied by strong subadditivity. We explicitly compute
its extremal rays for n up to 5. We also consider the symmetric monotonicity
cone, in which the formulas are required to be invariant under subsystem
permutations. We describe this cone fully for all n. In addition, we show that
these results hold even when states and operations are constrained to be
classical.
