\(L_ p\) affine isoperimetric inequalities.
type: Item
instance of: scholarly article
title: \(L_ p\) affine isoperimetric inequalities.
zbMATH Open document ID: 1034.52009
author: Erwin Lutwak
author: Deane Yang
author: Gaoyong Zhang
published in: Journal of Differential Geometry
publication date: 2002-08-15T00:00:00Z
review text: An affine isoperimetric inequality compares two functionals associated with convex (or more general) bodies, where the ratio of the functionals is invariant under non-degenerate linear transformations. The article deals with affine isoperimetric inequalities for centroid and projection bodies. The most important inequality concerning centroid bodies is the Busemann-Petty centroid inequality that answers positively a conjecture by Blaschke proving that the ratio of the volume of a body to that of its centroid body attains its maximum precisely for ellipsoids. The fundamental inequality for projection bodies is the Petty projection inequality: of all convex bodies of fixed volume, the ones whose polar projection bodies have maximal value are precisely the ellipsoids. Both the Petty projection inequality and the Busemann-Petty centroid inequality have come to be recognized as fundamental affine inequalities and have attracted increased attention. In this paper the authors derive the exact \(L_p\)-analogues of both the Busemann-Petty centroid inequality and the Petty projection inequality (as well as their equality conditions). The inequality concerning the \(L_p\)-centroid body is stronger than the well-known Blaschke-Santaló's inequality.
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A39
Mathematics Subject Classification ID: 52A21
zbMATH DE Number: 1782664
zbMATH Keywords: affine isoperimetric inequalities
zbMATH Keywords: centroid body
zbMATH Keywords: projection body
zbMATH Keywords: \(L_p\)-space
zbMATH Keywords: Busemann-Petty centroid inequality
zbMATH Keywords: Petty projection inequality
Wikidata QID: Q115203042
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1090347527
OpenAlex ID: W2148553975
DOI: 10.4310/JDG/1090347527
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description: scientific article; zbMATH DE number 1782664