Superadditivity of Fisher's information and logarithmic Sobolev inequalities

Abstract

We prove a theorem characterizing Gaussian functions and we prove a strict superaddivity property of the Fisher information. We use these results to determine the cases of equality in the logarithmic Sobolev inequality on R n equipped with Lebesgue measure and with Gauss measure. We also prove a strengthened form of Gross's logarithmic Sobolev inequality with a “remainder term” added to the left side. Finally we show that the strict form of Gross's inequality is a direct consequence of an inequality due to Blachman and Stam, and that this in turn is a direct consequence of strict superadditivity of the Fisher information.