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

Volume 213 / 2012

Keith Ball, Van Hoang Nguyen Studia Mathematica 213 (2012), 81-96 MSC: Primary 94A17. DOI: 10.4064/sm213-1-6

Abstract

We prove a quantitative dimension-free bound in the Shannon–Stam entropy inequality for the convolution of two log-concave distributions in dimension $d$ 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).

Authors

  • Keith BallInstitute of Mathematics
    University of Warwick
    Coventry, CV4 7AL, UK
    e-mail
  • Van Hoang NguyenInstitut de Mathématiques de Jussieu
    UPMC
    4 place Jussieu
    75252 Paris, France
    e-mail

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