The deficit in the Gaussian log-Sobolev inequality and inverse Santalo inequalities

Abstract

We establish dual equivalent forms involving relative entropy, Fisher information and optimal transport costs of inverse Santal{ó} 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{ó} inequalities some sharp lower bounds on the deficit in the Gaussian logarithmic Sobolev inequality.

References in corpus (5)

• A direct proof of the functional Santalo inequality
• Stein kernels and moment maps
• Partitions and functional Santalo inequalities
• Moment measures and stability for Gaussian inequalities
• Deficit Estimates for the Logarithmic Sobolev Inequality

Cited by in corpus (1)

• Stability of the Prekopa-Leindler inequality for log-concave functions