### Abstract

We introduce for p > - 1 the radial pth mean body RpK of a convex body K in double-struck^{n}. The distance from the origin to the boundary of RpK in a given direction is the pth mean of the distances from points inside K to the boundary of K in the same direction. The bodies RpK form a spectrum linking the difference body of K and the polar projection body of K, which correspond to p = ∞ and p = -1, respectively. We prove that RpK is convex when p > 0. We also establish a strong and sharp affine inequality relating the volume of RpK to that of RqK when -1 < p < q. When p = n and q → ∞, this becomes the Rogers-Shephard inequality, and when p → -1 and q = n, it becomes a reverse form of the Petty projection inequality proved previously by the second author.

Original language | English (US) |
---|---|

Pages (from-to) | 505-528 |

Number of pages | 24 |

Journal | American Journal of Mathematics |

Volume | 120 |

Issue number | 3 |

State | Published - Jun 1998 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*American Journal of Mathematics*,

*120*(3), 505-528.

**Affine inequalities and radial mean bodies.** / Gardner, R. J.; Zhang, Gaoyong.

Research output: Contribution to journal › Article

*American Journal of Mathematics*, vol. 120, no. 3, pp. 505-528.

}

TY - JOUR

T1 - Affine inequalities and radial mean bodies

AU - Gardner, R. J.

AU - Zhang, Gaoyong

PY - 1998/6

Y1 - 1998/6

N2 - We introduce for p > - 1 the radial pth mean body RpK of a convex body K in double-struckn. The distance from the origin to the boundary of RpK in a given direction is the pth mean of the distances from points inside K to the boundary of K in the same direction. The bodies RpK form a spectrum linking the difference body of K and the polar projection body of K, which correspond to p = ∞ and p = -1, respectively. We prove that RpK is convex when p > 0. We also establish a strong and sharp affine inequality relating the volume of RpK to that of RqK when -1 < p < q. When p = n and q → ∞, this becomes the Rogers-Shephard inequality, and when p → -1 and q = n, it becomes a reverse form of the Petty projection inequality proved previously by the second author.

AB - We introduce for p > - 1 the radial pth mean body RpK of a convex body K in double-struckn. The distance from the origin to the boundary of RpK in a given direction is the pth mean of the distances from points inside K to the boundary of K in the same direction. The bodies RpK form a spectrum linking the difference body of K and the polar projection body of K, which correspond to p = ∞ and p = -1, respectively. We prove that RpK is convex when p > 0. We also establish a strong and sharp affine inequality relating the volume of RpK to that of RqK when -1 < p < q. When p = n and q → ∞, this becomes the Rogers-Shephard inequality, and when p → -1 and q = n, it becomes a reverse form of the Petty projection inequality proved previously by the second author.

UR - http://www.scopus.com/inward/record.url?scp=0005899134&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0005899134&partnerID=8YFLogxK

M3 - Article

VL - 120

SP - 505

EP - 528

JO - American Journal of Mathematics

JF - American Journal of Mathematics

SN - 0002-9327

IS - 3

ER -