  In the world of generalized entropies---which, for example, play a role in
physical systems with sub- and super-exponential phasespace growth per degree
of freedom---there are two ways for implementing constraints in the maximum
entropy principle: linear- and escort constraints. Both appear naturally in
different contexts. Linear constraints appear e.g. in physical systems, when
additional information about the system is available through higher moments.
Escort distributions appear naturally in the context of multifractals and
information geometry. It was shown recently that there exists a fundamental
duality that relates both approaches on the basis of the corresponding deformed
logarithms (deformed-log duality). Here we show that there exists another
duality that arises in the context of information geometry, relating the Fisher
information of $\phi$-deformed exponential families that correspond to linear
constraints (as studied by J. Naudts), with those that are based on escort
constraints (as studied by S.-I. Amari). We explicitly demonstrate this
information geometric duality for the case of $(c,d)$-entropy that covers all
situations that are compatible with the first three Shannon-Khinchin axioms,
and that include Shannon, Tsallis, Anteneodo-Plastino entropy, and many more as
special cases. Finally, we discuss the relation between the deformed-log
duality and the information geometric duality, and mention that the escort
distributions arising in the two dualities are generally different and only
coincide for the case of the Tsallis deformation.
