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

Abstract We establish dual equivalent forms involving relative entropy, Fisher information, and optimal transport costs of inverse Santaló inequalities. show in particular that the Mahler conjecture is to some dimensional lower bound on deficit Gaussian logarithmic Sobolev inequality. also derive from existing results inequalities sharp bounds Our proofs rely duality relations between convex functionals (introduced [ 16] 62]) related notion moment measure.