Geodesic
Entropy jumps for isotropic log-concave random vectors and spectral gap
Keith Ball 1   ; Van Hoang Nguyen 2
1 Institute of Mathematics University of Warwick Coventry, CV4 7AL, UK
2 Institut de Mathématiques de Jussieu UPMC 4 place Jussieu 75252 Paris, France
Studia Mathematica, Tome 213 (2012) no. 1, pp. 81-96 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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).
DOI : 10.4064/sm213-1-6
Keywords: prove quantitative dimension free bound shannon stam entropy inequality convolution log concave distributions dimension terms spectral gap density method relies analysis fisher information production which second derivative entropy along normalized heat semigroup discuss consequences result study isotropic constant log concave distributions slicing problem

Keith Ball  1   ; Van Hoang Nguyen  2

1 Institute of Mathematics University of Warwick Coventry, CV4 7AL, UK
2 Institut de Mathématiques de Jussieu UPMC 4 place Jussieu 75252 Paris, France 
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Keith Ball; Van Hoang Nguyen. Entropy jumps for
 isotropic log-concave random vectors and spectral gap. Studia Mathematica, Tome 213 (2012) no. 1, pp. 81-96. doi: 10.4064/sm213-1-6

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