We present a new duality theory in order to treat convex optimization problems and we prove that the geometric duality used by C.H. Scott and T.R. Jefferson in different papers during the last quarter of century is a special case of it. Moreover, weaker sufficient conditions in order to achieve strong duality are considered and optimality conditions are derived. Next we apply our approach to some problems considered by Scott and Jefferson, determining their duals. We give weaker sufficient conditions in order to achieve strong duality and the corresponding optimality conditions. Finally, posynomial geometric programming is viewed also as a particular case of the duality approach we present.