Stability of inequalities in the dual Brunn-Minkowski theory
type: Item
instance of: scholarly article
title: Stability of inequalities in the dual Brunn-Minkowski theory
zbMATH Open document ID: 0926.52010
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-05-26T00:00:00Z
review text: The dual Brunn-Minkowski theory, initiated by Erwin Lutwak, has a series of inequalities which can be considered as dual counterparts to classical inequalities on mixed volumes and related functionals. A central topic of the dual theory are inequalities for integrals involving radial functions of star bodies. In the present paper, the authors obtain strong stability versions of some of these inequalities. These include the dual Brunn-Minkowski inequality, the dual Aleksandrov-Fenchel inequality, and the dual isoperimetric inequality. The proofs involve, among other tools, strong forms of Clarkson's inequality for \(L^p\) norms, and a refinement of the arithmetic-geometric mean inequality due to Kober. Finally, the authors give a more informative proof for the equivalence of the dual Brunn-Minkowski inequality and the dual Minkowski inequality.
reviewed by: Rolf Schneider
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 1274907
zbMATH Keywords: stability of geometric inequalities
zbMATH Keywords: star bodies
zbMATH Keywords: dual Brunn-Minkowski theory
author: Richard J. Gardner
author: Salvatore Vassallo
MaRDI profile type: Publication
full work available at URL: jmaa.1998.6254
OpenAlex ID: W1968420523
cites work: Intersection Bodies and the Busemann-Petty Problem
cites work: A positive answer to the Busemann-Petty problem in three dimensions
cites work: Q4856168
cites work: Geometric tomography.
cites work: Inequalities for dual isoperimetric deficits
cites work: Tomography of convex and star bodies
cites work: Stability Results for First Order Projection Bodies
cites work: On the Brunn-Minkowski theorem
cites work: On an inequality of Minkowski for mixed volumes
cites work: Q4274001
cites work: Q4064235
cites work: Q3908030
cites work: On the Arithmetic and Geometric Means and on Holder's Inequality
cites work: Dual mixed volumes
cites work: Analytic Inequalities
cites work: Q5287501
cites work: Stability in the Aleksandrov‐Fenchel‐Jessen Theorem
cites work: A stability estimate for the Aleksandrov-Fenchel inequality, with an application to mean curvature
cites work: Q5287551
cites work: \(L_p\) metrics for compact convex sets
cites work: A positive solution to the Busemann-Petty problem in \(\mathbb{R}^4\)
cites work: Intersection bodies and the Busemann-Petty inequalities in \(\mathbb{R}^ 4\)
DOI: 10.1006/JMAA.1998.6254
description: scientific article; zbMATH DE number 1274907