  We investigate the problem of optimally approximating a desired state by the
convex mixing of a set of available states. The problem is recasted as finding
the optimal state with the minimum distance from target state in a convex set
of usable states. Based on the fidelity, we define the optimal convex
approximation of an expected state and present the complete exact solutions
with respect to an arbitrary qubit state. We find that the optimal state based
on fidelity is closer to the target state than the optimal state based on trace
norm in many ranges. Finally, we analyze the geometrical properties of the
target states which can be completely represented by a set of practicable
states. Using the feature of convex combination, we express this class of
target states in terms of three available states.
