  We present a convexity-type result concerning simple quasi-states on closed
manifolds. As a corollary, an inequality emerges which relates the Poisson
bracket and the measure of non-additivity of a simple quasi-state on a closed
surface equipped with an area form. In addition, we prove that the uniform norm
of the Poisson bracket of two functions on a surface is stable from below under
C^0-perturbations.
