Brunn-Minkowski inequality for mixed intersection bodies
type: Item
instance of: scholarly article
title: Brunn-Minkowski inequality for mixed intersection bodies
zbMATH Open document ID: 1065.52006
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-01-26T00:00:00Z
review text: In 1985 Lutwak introduced the notion of mixed projection bodies and obtained the Brunn-Minkowski inequality for these bodies. In this paper the authors prove the corresponding inequality for mixed intersection bodies. Besides its intrinsic interest this result is also an interesting example showing the duality between projection and intersection bodies. In particular they prove the following results: Let \(x_1 \tilde + \cdots \tilde + x_r\) denote the usual vector sum of \(x_1, \dots , x_r\) if all \(x_1, \dots , x_r\) lie in the same \(1\)-dimensional subspace, and \(0\) otherwise. If \(K_1=\dots =K_{n-i}=K\) and \(K_{n-i+1}=\dots =K_n=L\) the dual mixed volumes are written as \(V_i(K,L)\). Let \(\widetilde W_i(K)\) denote the dual mixed volume \(\widetilde V_i(K,B)\). Let \(IK\) denote the intersection body of \(K\), and \(I_iK\) the \(i\)th mixed intersection body of \(K\). If \(K, L\) are \(n\)-dimensional star-shaped bodies, then for \(0 \leq i < n\), \[ \widetilde W_i(I(K\tilde + L))^{\frac1{(n-i)(n-1)}}\leq \widetilde W_i (IK)^{\frac1{(n-i)(n-1)}} + \widetilde W_i (IL)^{\frac1{(n-i)(n-1)}}, \] with equality if and only if \(K\) and \(L\) are dilates. They also obtain the following more general statement: If \(K, L\) are \(n\)-dimensional star-shaped bodies, then for \(0 \leq i < n\), and for \(0 \leq j <n-2\), \[ \widetilde W_i(I_j(K\tilde + L))^{\frac1{(n-i)(n-j-1)}}\leq \widetilde W_i (I_jK)^{\frac1{(n-i)(n-j-1)}} + \widetilde W_i (I_jL)^{\frac1{(n-i)(n-j-1)}}, \] with equality if and only if \(K\) and \(L\) are dilates.
Mathematics Subject Classification ID: 52A39
zbMATH DE Number: 2131162
zbMATH Keywords: dual mixed volumes
zbMATH Keywords: intersection bodies
zbMATH Keywords: mixed intersection bodies
author: Changjian Zhao
author: Gangsong Leng
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.07.013
OpenAlex ID: W2092225209
cites work: Spherical projections and centrally symmetric sets
cites work: A positive answer to the Busemann-Petty problem in three dimensions
cites work: Q4856168
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 dual mixed volumes
cites work: Inequalities for Mixed Projection Bodies
cites work: Volume of Mixed Bodies
cites work: Mixed Projection Inequalities
cites work: Dual mixed volumes
cites work: Centered Bodies and Dual Mixed Volumes
cites work: A positive solution to the Busemann-Petty problem in \(\mathbb{R}^4\)
DOI: 10.1016/J.JMAA.2004.07.013
description: scientific article; zbMATH DE number 2131162