  In this paper, we examine the superadditivity of convex roof coherence
measures. We put forward a theorem on the superadditivity of convex roof
coherence measures, which provides a sufficient condition to identify the
convex roof coherence measures fulfilling the superadditivity. By applying the
theorem to each of the known convex roof coherence measures, we prove that the
coherence of formation and the coherence concurrence are superadditive, while
the geometric measure of coherence, the convex roof coherence measure based on
linear entropy, the convex roof coherence measure based on fidelity, and convex
roof coherence measure based on $\frac{1}{2}$-entropy are non-superadditive.
