Document Type

Article

Publication Date

Summer 1999

Abstract

If K is a convex body in En, its cross-section body CK has a radial function in any direction u is ∈ Sn-1 equal to the maximal volume of hyperplane sections of K orthogonal to u. A generalization called the p-cross-section body CpK of K, where p > -1, is introduced. The radial function of CpK in any direction u ∈ Sn-1 is the pth mean of the volumes of hyperplane sections of K orthogonal to u through points in K. It is shown that C1K is convex but CpK is generally not convex when p > 1. An inclusion of the form an,qCqK ⊆ an,pCpK, where -1 < p < q and the constant an,p is the best possible, is established. This is applied to disprove a conjecture of Makai and Martini.

Publication Title

Indiana University Mathematics Journal

Volume

48

Issue

2

First Page

593

Last Page

613

Subjects - Topical (LCSH)

Convex bodies; Radial basis functions

Genre/Form

articles

Type

Text

Rights

Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.

Language

English

Format

application/pdf

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Included in

Mathematics Commons

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