Mathematika

Research Article

On the reverse Lp–busemann–petty centroid inequality

Stefano Campia1 and Paolo Gronchia2

a1 Dipartimento di Matematica Pura e Applicata “G. Vitali”, Università degli Studi di Modena e Reggio Emilia, Via Campi 213 B, 41100 Modena, Italy. E-mail: campi@unimo.it

a2 Istituto per le Applicazioni del Calcolo, Consiglio Nazionale delle Ricerche, Via Madonna del Piano- Edificio F, 50019 Sesto Fiorentino (FI), Italy. E-mail: paolo@fi.iac.cnr.it

Abstract

The volume of the Lp-centroid body of a convex body K d is a convex function of a time-like parameter when each chord of K parallel to a fixed direction moves with constant speed. This fact is used to study extrema of some affine invariant functionals involving the volume of the Lp-centroid body and related to classical open problems like the slicing problem. Some variants of the Lp-Busemann-Petty centroid inequality are established. The reverse form of these inequalities is proved in the two-dimensional case.

(Received February 01 2002)

Key Words:

  • 52A40: CONVEX AND DISCRETE GEOMETRY; General convexity; Inequalities and extremum problems.