  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.
