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

Abstract

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).

15 pages, The title is changed and Acknowledgement is added

References in corpus (2)

• The entropy per coordinate of a random vector is highly constrained under convexity conditions
• Dimensional behaviour of entropy and information

Cited by in corpus (1)

• Entropy and the fourth moment phenomenon