Testing statistical composite hypotheses is a very difficult area of the mathematical statistics theory and optimal solutions are found in very seldom cases. It is precisely in this respect that the paper ''Hypotheses testing by convex optimization'' brings a new insight and a powerful contribution. The optimality of solutions depends strongly on the criterion adopted for measuring the risk of a statistical procedure. In our opinion, the novelty here lies in the introduction of a new criterion different from the usual one. In the present discussion, we give some more precise details on the main results necessary to enlighten the strength and the limits of the new theory.