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Stability of the entropy for superstatistics

The Boltzmann-Gibbs celebrated entropy $S_{BG}=-k\sum_ip_i \ln p_i$ is {\it concave} (with regard to all probability distributions $\{p_i\}$) and {\it stable} (under arbitrarily small deformations of any given probability distribution). It seems reasonable to consider these two properties as {\it necessary} for an entropic form to be a {\it physical} one in the thermostatistical sense. Most known entropic forms (e.g., Renyi entropy) violate these conditions, in contrast with the basis of nonextensive statistical mechanics, namely $S_q=k\frac{1-\sum_ip_i^q}{q-1} (q\in {\cal R}; S_1=S_{BG})$, which satisfies both ($\forall q>0$). We have recently generalized $S_q$ (into $S$) in order to yield, through optimization, the Beck-Cohen superstatistics. We show here that $S$ satisfies both conditions as well. Given the fact that the (experimentally observed) optimizing distributions are invariant through {\it any} monotonic function of the entropic form to be optimized, this might constitute a very strong criterion for identifying the physically correct entropy.