Affine inequalities and radial mean bodies

R. J. Gardner, Gaoyong Zhang

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Pages (from-to)505-528
Number of pages24
JournalAmerican Journal of Mathematics
Volume120
Issue number3
StatePublished - Jun 1998

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

In: American Journal of Mathematics, Vol. 120, No. 3, 06.1998, p. 505-528.

Research output: Contribution to journalArticle

Gardner, R. J. ; Zhang, Gaoyong. / Affine inequalities and radial mean bodies. In: American Journal of Mathematics. 1998 ; Vol. 120, No. 3. pp. 505-528.
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