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.

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U2 - 10.1353/ajm.1998.0021

DO - 10.1353/ajm.1998.0021

M3 - Article

AN - SCOPUS:0005899134

SN - 0002-9327

VL - 120

SP - 505

EP - 528

JO - American Journal of Mathematics

JF - American Journal of Mathematics

IS - 3

ER -