Entropy jumps for isotropic log-concave random vectors and spectral gap

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).