It is shown that if X is a random variable whose density satisfies a Poincaré inequality, and Y is an independent copy of X, then the entropy of (X + Y)/√2 is greater than that of X by a fixed fraction of the entropy gap between X and the Gaussian of the same variance. The argument uses a new formula for the Fisher information of a marginal, which can be viewed as a local, reverse form of the Brunn-Minkowski inequality (in its functional form due to A. Prékopa and L. Leindler).
"Entropy jumps in the presence of a spectral gap." Duke Math. J. 119 (1) 41 - 63, 15 July 2003. https://doi.org/10.1215/S0012-7094-03-11912-2