#### Title

#### Document Type

Article

#### Publication Date

Summer 1999

#### Abstract

If *K* is a convex body in **E**^{n}, its cross-section body *CK* has a radial function in any direction *u* is ∈ *S ^{n}*

^{-1}equal to the maximal volume of hyperplane sections of

*K*orthogonal to

*u*. A generalization called the

*p*-cross-section body

*C*of

_{p}K*K*, where

*p*> -1, is introduced. The radial function of

*C*in any direction

_{p}K*u*∈

*S*

^{n-}^{1}is the

*p*th mean of the volumes of hyperplane sections of

*K*orthogonal to

*u*through points in

*K*. It is shown that

*C*1

*K*is convex but

*C*is generally not convex when

_{p}K*p*> 1. An inclusion of the form

*a*⊆

_{n,q}C_{q}K*a*, where -1 <

_{n,p}C_{p}K*p*<

*q*and the constant

*a*is the best possible, is established. This is applied to disprove a conjecture of Makai and Martini.

_{n,p}#### Publication Title

Indiana University Mathematics Journal

#### Volume

48

#### Issue

2

#### First Page

593

#### Last Page

613

#### Recommended Citation

Gardner, Richard J. and Giannopoulos, A. A., "p-Cross-section Bodies" (1999). *Mathematics.* Paper 23.

http://cedar.wwu.edu/math_facpubs/23