  We establish dual equivalent forms involving relative entropy, Fisher
information and optimal transport costs of inverse Santal{\'o} inequalities. We
show in particular that the Mahler conjecture is equivalent to some dimensional
lower bound on the deficit in the Gaussian logarithmic Sobolev inequality. We
also derive from existing results on inverse Santal{\'o} inequalities some
sharp lower bounds on the deficit in the Gaussian logarithmic Sobolev
inequality.
