## Abstract

We show that the fundamental objects of the L _{p}-Brunn-Minkowski theory, namely the L _{p}-affine surface areas for a convex body, are closely related to information theory: they are exponentials of Rényi divergences of the cone measures of a convex body and its polar. We give geometric interpretations for all Rényi divergences D _{α}, not just for the previously treated special case of relative entropy which is the case α=1. Now, no symmetry assumptions are needed and, if at all, only very weak regularity assumptions are required. Previously, the relative entropies appeared only after performing second order expansions of certain expressions. Now already first order expansions make them appear. Thus, in the new approach we detect "faster" details about the boundary of a convex body.

Original language | English (US) |
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Pages (from-to) | 1040-1059 |

Number of pages | 20 |

Journal | Advances in Mathematics |

Volume | 230 |

Issue number | 3 |

DOIs | |

State | Published - Jun 20 2012 |

Externally published | Yes |

### Bibliographical note

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

## Keywords

- L -affine surface area
- Relative entropy
- Rényi divergence

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