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

2020/07/10 by Nathaël Gozlan, Gozlan, Nathaël
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Optimization and Control (math.OC) #Point processes and geometric inequalities #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2007.05255

openalex publication_date 2020/07/10 · openalex created_date 2020/07/16 · openalex updated_date 2026/07/28

Abstract

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

Citations

• Logarithmic Sobolev Inequalities
• Deficit Estimates for the Logarithmic Sobolev Inequality

Related

• Bounds on the deficit in the logarithmic Sobolev inequality
• Remark on the stability of Log-Sobolev inequality for Gaussian measure
• Quantitative logarithmic Sobolev inequalities and stability estimates
• Transport proofs of some functional inverse Santaló inequalities
• Stein's method, logarithmic Sobolev and transport inequalities