The plank problem for symmetric bodies.
Logarithmically concave functions and sections of convex sets in
Sharp uniform convexity and smoothness inequalities for trace norms.
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 in terms 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 semigroup. We also discuss consequences of our result in the study of the isotropic constant of log-concave distributions (slicing problem).
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