## Abstract

A number of sharp geometric inequalities for polars of mixed projection bodies (zonoids) are obtained. Among the inequalities derived is a polar projection inequality that has the projection inequality of Petty as a special case. Other special cases of this polar projection inequality are inequalities (between the volume of a convex body and that of the polar of its 2th projection body) that are strengthened forms of the classical inequalities between the volume of a convex body and its projection measures (Quermassintegrale). The relation between the Busemann-Petty centroid inequality and the Petty projection inequality is shown to be similar to the relation that exists between the Blaschke-Santalo inequality and the affine isoperimetric inequality of affine differential geometry. Some mixed integral inequalities are derived similar in spirit to inequalities obtained by Chakerian and others.

Original language | English (US) |
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Pages (from-to) | 91-106 |

Number of pages | 16 |

Journal | Transactions of the American Mathematical Society |

Volume | 287 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1985 |

## Keywords

- Centroid body
- Convex body
- Mixed area measure
- Mixed volume
- Projection measure (Quermassintegral)
- Projection body
- Zonoid

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics