### 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) |
---|---|

Pages (from-to) | 1811-1820 |

Number of pages | 10 |

Journal | Transactions of the American Mathematical Society |

Volume | 349 |

Issue number | 5 |

State | Published - 1997 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Transactions of the American Mathematical Society*,

*349*(5), 1811-1820.

**Bodies with similar projections.** / Chakerian, G. D.; Lutwak, B.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 349, no. 5, pp. 1811-1820.

}

TY - JOUR

T1 - Bodies with similar projections

AU - Chakerian, G. D.

AU - Lutwak, B.

PY - 1997

Y1 - 1997

N2 - 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.

AB - 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.

KW - Convex body

KW - Generalized zonoid

KW - Mixed volume

KW - Quermassintegral

KW - Relative brightness

KW - Relative girth

KW - Zonoid

UR - http://www.scopus.com/inward/record.url?scp=21744436022&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21744436022&partnerID=8YFLogxK

M3 - Article

VL - 349

SP - 1811

EP - 1820

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 5

ER -