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We present a convexity-type result concerning simple quasi-states on
closed manifolds. As a corollary, an inequality emerges which relates
the Poisson bracket to 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.