Rearrangements and information theoretic inequalities

Abstract

We investigate the interaction of functional rearrangements with information theoretic inequalities. In particular we will prove the Relative Fisher information from Gaussianity decreases on half-space rearrangement, as a consequence we get a qualitative sharpening of the usual Gaussian log-Sobolev inequality. Additionally, we compare this half space rearrangement’s interaction with distance from Gaussianity, with the spherical rearrangement’s role in entropy power inequalities.