Convexity properties of superpositions of degenerate bipartite eigenstates

The entanglement content of superpositions of pairs of degenerate eigenstates of a bipartite system is considered in the case that both eigenstates are also eigenstates of the $z$ component of the total angular momentum. It is shown that the von Neumann entropy of the state that is obtained tracing out one of the parts of the system has a definite convexity (concavity) as a function of the superposition parameter and that its convexity (concavity) can be predicted using a quantity of information that measures the entropy shared by the states at the extremes of the superposition. Several examples of two-particle systems, whose eigenfunctions and density matrices can be obtained exactly, are analyzed thoroughly.