Entropy Maximization with Linear Constraints: The Uniqueness of the Shannon Entropy

Abstract: Within a framework of utmost generality, we show that the entropy maximization procedure with linear constraints uniquely leads to the Shannon-Boltzmann-Gibbs entropy. Therefore, the use of this procedure with linear constraints should not be extended to the generalized entropies introduced recently. In passing, it is remarked how the forceful use of the entropy maximization for the Tsallis and R\'enyi entropies implies either the Shannon limit of these entropies or self-referential contradictions. Finally, we note that the utilization of the entropy maximization procedure with different averaging schemes is beyond the scope of this work.