Some inequalities about dual mixed volumes of star bodies
type: Item
instance of: scholarly article
title: Some inequalities about dual mixed volumes of star bodies
zbMATH Open document ID: 1094.52005
DOI: 10.1016/S0252-9602(05)60013-6
published in: Acta Mathematica Scientia. Series B. (English Edition)
publication date: 2005-10-18T00:00:00Z
review text: The authors present some inequalities involving dual mixed volumes of star bodies. These inequalities are analogous to the corresponding ones for the quermassintegrals of a convex body. There are interesting new examples on the duality existing in Geometric Tomography between projections and sections. Let \(K\) and \(L\) be an arbitrary pair of star bodies in the \(n\)-dimensional Euclidean space; let \(B\) denote the unit ball; then \[ \frac {\Tilde V_k(K \Tilde + L)}{\Tilde V_{k-1}(K \Tilde + L)} \leq \frac {\Tilde V_k(K)}{\Tilde V_{k-1}(K)} + \frac {\Tilde V_k(L)}{\Tilde V_{k-1}(L)} \] for \(k=1,2\), where \(\Tilde V_i(K)\) is the \(i\)th coefficient in volume of the radial Minkowski linear combination of \(i\) copies of \(K\) and \((n-i)\) copies of \(B\). Let \(K\) be a star body and \(L\) a ball in the \(n\)-dimensional Euclidean space; then the inequality above holds for \(k=1,\dots,n\).
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A39
Mathematics Subject Classification ID: 52A30
zbMATH DE Number: 2215148
zbMATH Keywords: Brunn-Minkowski theory
zbMATH Keywords: star bodies
zbMATH Keywords: dual mixed volumes
zbMATH Keywords: Aleksandrov-Fenchel inequality
author: Xiaoyan Li
author: Gangsong Leng
MaRDI profile type: Publication
full work available at URL: s0252-9602(05)60013-6
OpenAlex ID: W2784097247
description: scientific article