  In this short and elementary note, we study some ergodic optimization
problems for circle expanding maps. We first make an observation that if a
function is not far from being convex, then its calibrated sub-actions are
closer to convex functions in certain effective way. As an application of this
simple observation, for circle doubling map, we generalize a result of Bousch
saying that translations of the cosine function are uniquely optimized by
Sturmian measures. Our argument follows the mainline of Bousch's original
proof, while the technical part is simplified by the observation mentioned
above, and no numerical calculation is needed.
