  State convertibility is fundamental in the study of resource theory of
quantum coherence. It is aimed at identifying when it is possible to convert a
given coherent state to another using only incoherent operations. In this
paper, we give a complete characterization of state convertibility under
genuinely incoherent operations. It is found that the convexity of the
robustness of coherence plays a central role. Based on this, the majorization
condition of determining convertibility from pure states to mixed states under
strictly incoherent operations is provided. Moreover, maximally coherent states
in the set of all states with fixed diagonal elements are determined. It is
somewhat surprising that the convexity of the robustness of coherence can also
decide conversion between off-diagonal parts of coherent states. This might be
a big step to answer completely the question of state convertibility for mixed
states under incoherent operations.
