Entropy jumps in the presence of a spectral gap (Q1409333)

Let \(X\) be a random variable whose density satisfies a Poincaré inequality and \(Y\) be an independent copy of \(X\). The authors show that the entropy of \((X + Y)/\sqrt 2\) is greater than that of \(X\) by a fixed fraction of the entropy gap between \(X\) and the Gaussian with the same variance.