  We prove a quantitative dimension-free bound in the Shannon-Stam Entropy
inequality for the convolution of two log-concave distributions in dimension d
interms of the spectral gap of the density. The method relies on the analysis
of the Fisher Information production, which is the second derivative of the
Entropy along the (normalized) Heat semi-group. We also discuss consequences of
our result in the study of the isotropic constant of log-concave distributions
(slicing problem).
