\(L_p\)-mixed intersection bodies and star duality
type: Item
instance of: scholarly article
title: \(L_p\)-mixed intersection bodies and star duality
zbMATH Open document ID: 1204.52012
author: Changjian Zhao
author: Wing-Sum Cheung
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 2010-12-14T00:00:00Z
review text: For star bodies \(K_1,\dots,K_n\) in \({\mathbb R}^n\), the authors introduce the \(q\)-dual mixed volume (\(q\not=0\)) by \[ \tilde V_q(K_1,\dots,K_n)=\omega_n\left(\frac{1}{n\omega_n} \int_{S^{n-1}} \rho^q(K_1,u)\cdots\rho^q(K_n,u)\, dS(u)\right)^{1/q}, \] where the notations are as usual (e.g., \(\rho\) is the radial function), and the \(L_p\)-mixed intersection body \({\mathbf I}_p(K_1, \dots, K_{n-1})\) (\(p<1\)) by \[ \rho^p({\mathbf I}_p(K_1,\dots,K_{n-1}),u) =\frac{2}{1-p}\tilde v_p(K_1\cap E_u,\dots,K_{n-1}E_u), \] where \(\tilde v_p\) is the \(p\)-dual mixed volume in \((n-1)\)-dimensional spaces. Various inequalities, involving these objects and the star duality \(K\mapsto K^\circ\), defined by \(\rho(K^\circ,\cdot)=1/\rho(K,\cdot)\), are obtained, mainly by applications of Hölder's integral inequality.
reviewed by: Rolf Schneider
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A30
zbMATH DE Number: 5826106
zbMATH Keywords: \(q\)-dual mixed volume
zbMATH Keywords: \(L_p\)-mixed intersection body
zbMATH Keywords: star duality
MaRDI profile type: Publication
full work available at URL: s12044-010-0041-0
OpenAlex ID: W1994429193
cites work: Volume in terms of concurrent cross-sections
cites work: Stability estimates for star bodies in terms of their intersection bodies
cites work: Convex intersection bodies in three and four dimensions
cites work: Spherical projections and centrally symmetric sets
cites work: On the Busemann-Petty problem\\ concerning central sections\\ of centrally symmetric convex bodies
cites work: Intersection Bodies and the Busemann-Petty Problem
cites work: A positive answer to the Busemann-Petty problem in three dimensions
cites work: An analytic solution to the Busemann-Petty problem
cites work: An analytic solution to the Busemann-Petty problem on sections of convex bodies
cites work: Functional analytic characterizations of classes of convex bodies
cites work: Intersection bodies and ellipsoids
cites work: Convolutions, Transforms, and Convex Bodies
cites work: A characterization of Lp intersection bodies
cites work: A functional analytic approach to intersection bodies
cites work: Intersection bodies, positive definite distributions, and the Busemann-Petty problem
cites work: Second derivative test for intersection bodies
cites work: Intersection bodies in \(\mathbb{R}^4\)
cites work: Intersection bodies and the Busemann-Petty problem
cites work: Intersection bodies and valuations
cites work: Intersection bodies and dual mixed volumes
cites work: Inequalities for Mixed Projection Bodies
cites work: Mixed Projection Inequalities
cites work: Dual mixed volumes
cites work: Quotient star bodies, intersection bodies, and star duality
cites work: Intersection bodies and polytopes
cites work: Sections of Convex Bodies
cites work: Intersection bodies and the Busemann-Petty inequalities in \(\mathbb{R}^ 4\)
cites work: Centered Bodies and Dual Mixed Volumes
cites work: Q4277373
cites work: A positive solution to the Busemann-Petty problem in \(\mathbb{R}^4\)
DOI: 10.1007/S12044-010-0041-0
Recommended article: On star duality of mixed intersection bodies
Recommended article: \(L_{p}\)-mixed intersection bodies
Recommended article: Quotient star bodies, intersection bodies, and star duality
Recommended article: INEQUALITIES FOR STAR DUALS OF INTERSECTION BODIES
Recommended article: Q3071181
Recommended article: The dual Brunn-Minkowski inequalities for star dual of mixed intersection bodies
Recommended article: \(q\)-dual mixed volumes and \(L_{p}\)-intersection bodies
Recommended article: \(L_p\) intersection bodies
description: scientific article; zbMATH DE number 5826106