## Abstract

We introduce a new class of (not necessarily convex) bodies and show, among other things, that these bodies provide yet another link between convex geometric analysis and information theory. Namely, they give geometric interpretations of the relative entropy of the cone measures of a convex body and its polar and related quantities. Such interpretations were first given by Paouris and Werner for symmetric convex bodies in the context of the L_{p}-centroid bodies. There, the relative entropies appear after performing second order expansions of certain expressions. Now, no symmetry assumptions are needed. Moreover, using the new bodies, already first order expansions make the relative entropies appear. Thus, these bodies detect “faster” details of the boundary of a convex body than the L_{p}-centroid bodies.

Original language | English (US) |
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Pages (from-to) | 2889-2906 |

Number of pages | 18 |

Journal | Transactions of the American Mathematical Society |

Volume | 366 |

Issue number | 6 |

DOIs | |

State | Published - 2014 |

### Bibliographical note

Publisher Copyright:© 2014 American Mathematical Society.

## Keywords

- L-affine surface area
- Mean width
- Relative entropy