### Abstract

Aleksandrov's projection theorem characterizes centrally symmetric convex bodies by the measures of their orthogonal projections on lower dimensional subspaces. A general result proved here concerning the mixed volumes of projections of a collection of convex bodies has the following corollary. If K is a convex body in R" whose projections on r-dimensional subspaces have the same r-dimensional volume as the projections of a centrally symmetric convex body A/, then the Quermassintegrals satisfy \Vj(M) Wj(K), for 0 < j < n -r, with equality, for any j, if and only if K is a translate of M. The case where K is centrally symmetric gives Aleksandrov's projection theorem.

Original language | English (US) |
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Pages (from-to) | 1811-1820 |

Number of pages | 10 |

Journal | Transactions of the American Mathematical Society |

Volume | 349 |

Issue number | 5 |

State | Published - Dec 1 1997 |

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### Keywords

- Convex body
- Generalized zonoid
- Mixed volume
- Quermassintegral
- Relative brightness
- Relative girth
- Zonoid

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*349*(5), 1811-1820.