## Abstract

Let K be a convex body in ℝ ^{n}. We introduce a new affine invariant, which we call Ω _{K}, that can be found in three different ways: as a limit of normalized L _{p}-affine surface areas;as the relative entropy of the cone measure of K and the cone measure of K°;as the limit of the volume difference of K ^{̊} and L _{p}-centroid bodies. We investigate properties of Ω _{K} and of related new invariant quantities. In particular, we show new affine isoperimetric inequalities and we show an 'information inequality' for convex bodies.

Original language | English (US) |
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Pages (from-to) | 253-286 |

Number of pages | 34 |

Journal | Proceedings of the London Mathematical Society |

Volume | 104 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2012 |

Externally published | Yes |

### Bibliographical note

Funding Information:The first author was partially supported by an NSF grant. The second author was partially supported by an NSF grant, an FRG-NSF grant and a BSF grant.

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