Rearrangements and information theoretic inequalities

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.