The Deficit in the Gaussian Log-Sobolev Inequality and Inverse Santalo Inequalities

We establish dual equivalent forms involving relative entropy, Fisher information, and optimal transport costs of inverse Santalo 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 Santalo inequalities some sharp lower bounds on the deficit in the Gaussian logarithmic Sobolev inequality. Our proofs rely on duality relations between convex functionals (introduced in [16] and [62]) related to the notion of moment measure.