The \(L^p\)-Busemann-Petty centroid inequality
type: Item
instance of: scholarly article
title: The \(L^p\)-Busemann-Petty centroid inequality
zbMATH Open document ID: 1002.52005
author: Stefano Campi
author: Paolo Gronchi
published in: Advances in Mathematics
publication date: 2002-07-04T00:00:00Z
review text: The authors consider parallel chord movements of convex bodies; in such movements the chords of a convex body parallel to a fixed direction are shifted during a time \(t\) -- each chord with different speed -- with the only condition that the union of the shifted chords remains a convex body. These movements are particular cases of shadow systems and include as a special case the movement related to Steiner symmetrization. The authors prove that the family of all the \(p\)-centroid bodies of the parallel chord movements of a convex body along a fixed direction \(v\) is a shadow system along the same direction \(v\). They also prove that the volume function of the \(p\)-centroid body in such parallel chord movements is a strictly convex function unless the speed function of the movement is linear. As a consequence of this result they give an alternative proof of a result by Lutwak, Yang and Zhang characterizing the origin symmetric ellipsoids as the minimizers of the function \(\frac {V(\Gamma_p K)}{V(K)}\) where \(\Gamma_p K\) is the \(p\)-centroid body of \(K\).
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 1763529
zbMATH Keywords: centroid body
zbMATH Keywords: Busemann-Petty centroid inequality
zbMATH Keywords: Steiner symmetrization
zbMATH Keywords: affine isoperimetric inequalities
zbMATH Keywords: shadow system
zbMATH Keywords: parallel chord movements
MaRDI profile type: Publication
full work available at URL: aima.2001.2036
OpenAlex ID: W2052546116
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DOI: 10.1006/AIMA.2001.2036
description: scientific article; zbMATH DE number 1763529