### Abstract

It is shown that if X is a random variable whose density satisfies a Poincaré inequality, and Y is an independent copy of X, then the entropy of (X + Y)/2√ is greater than that of X by a fixed fraction of the entropy gap between X and the Gaussian of the same variance. The argument uses a new formula for the Fisher information of a marginal, which can be viewed as a local, reverse form of the Brunn-Minkowski ineauality (in its functional form due to A. Prékopa and L. Leindler).

Original language | English (US) |
---|---|

Pages (from-to) | 41-63 |

Number of pages | 23 |

Journal | Duke Mathematical Journal |

Volume | 119 |

Issue number | 1 |

DOIs | |

State | Published - Jul 15 2003 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Duke Mathematical Journal*,

*119*(1), 41-63. https://doi.org/10.1215/S0012-7094-03-11912-2

**Entropy jumps in the presence of a spectral gap.** / Ball, Keith; Barthe, Franck; Naor, Assf.

Research output: Contribution to journal › Article

*Duke Mathematical Journal*, vol. 119, no. 1, pp. 41-63. https://doi.org/10.1215/S0012-7094-03-11912-2

}

TY - JOUR

T1 - Entropy jumps in the presence of a spectral gap

AU - Ball, Keith

AU - Barthe, Franck

AU - Naor, Assf

PY - 2003/7/15

Y1 - 2003/7/15

N2 - It is shown that if X is a random variable whose density satisfies a Poincaré inequality, and Y is an independent copy of X, then the entropy of (X + Y)/2√ is greater than that of X by a fixed fraction of the entropy gap between X and the Gaussian of the same variance. The argument uses a new formula for the Fisher information of a marginal, which can be viewed as a local, reverse form of the Brunn-Minkowski ineauality (in its functional form due to A. Prékopa and L. Leindler).

AB - It is shown that if X is a random variable whose density satisfies a Poincaré inequality, and Y is an independent copy of X, then the entropy of (X + Y)/2√ is greater than that of X by a fixed fraction of the entropy gap between X and the Gaussian of the same variance. The argument uses a new formula for the Fisher information of a marginal, which can be viewed as a local, reverse form of the Brunn-Minkowski ineauality (in its functional form due to A. Prékopa and L. Leindler).

UR - http://www.scopus.com/inward/record.url?scp=0042236601&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0042236601&partnerID=8YFLogxK

U2 - 10.1215/S0012-7094-03-11912-2

DO - 10.1215/S0012-7094-03-11912-2

M3 - Article

VL - 119

SP - 41

EP - 63

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 1

ER -