A geometric inequality on mixed volumes
type: Item
instance of: scholarly article
title: A geometric inequality on mixed volumes
zbMATH Open document ID: 0873.52007
DOI: 10.1007/BF00129897
published in: Annals of Global Analysis and Geometry
publication date: 1997-10-26T00:00:00Z
review text: For convex bodies \(K,M\) in \(\mathbb{R}^n\), the inequality \[ V_{n- 1}(K,M)\leq \rho^{n-1}W_{n-1}(K) \tag{\(*\)} \] (where \(V_{n-1}(K,M)= V(K,M,\dots,M)\) denotes the mixed volume, \(W_{n-1}(K)\) the \((n-1)\)st quermassintegral, and \(\rho\) is the circumradius of \(M\)) follows directly from the monotonicity of mixed volumes. The author extends \((*)\) to quasiconvex sets \(W\) (these are compact sets with smooth boundary which have convex hyperplane sections in a certain direction). The extension states that \[ V_{n-1}(c(W),M)\leq {\textstyle {\frac1n}} \rho^{n-1}M_{n-2}(W), \] where \(c(W)\) is the convex hull and \(M_{n- 2}(W)\) the \((n-2)\)th curvature integral. The proof is based on a connectivity result for quasi-convex sets and on the principal kinematic formula.
Mathematics Subject Classification ID: 52A30
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A22
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 955340
zbMATH Keywords: mixed volume
zbMATH Keywords: quasiconvex sets
zbMATH Keywords: curvature integral
author: Y. D. Chai
reviewed by: Wolfgang Weil
MaRDI profile type: Publication
cites work: Q4753788
cites work: Q5593766
cites work: A topological characterization of compact \(n\)-manifolds with \(C^ 2\)- boundary in \(R^ n\)
cites work: On the Kinematic Formula in the Euclidean Space of N Dimensions
cites work: The mixed area of a convex body and its polar reciprocal
cites work: A Proof of Minkowski's Inequality for Convex Curves
cites work: Polar Duals of Convex Bodies
cites work: Bonnesen-Style Isoperimetric Inequalities
cites work: Isoperimetric Inequalities in Mathematical Physics. (AM-27)
cites work: Q5287551
Recommended article: A geometric proof of some inequalities involving mixed volumes
Recommended article: On an inequality of Minkowski for mixed volumes
Recommended article: Some inequalities about mixed volumes
Recommended article: Convex Bodies: Mixed Volumes and Inequalities
Recommended article: Q3014811
Recommended article: Bezout Inequality for Mixed Volumes
Recommended article: Q5702126
Recommended article: INEQUALITIES BETWEEN MIXED VOLUMES OF CONVEX BODIES: VOLUME BOUNDS FOR THE MINKOWSKI SUM
Recommended article: Some inequalities for \((p,q)\)-mixed volume
Recommended article: A Volume Inequality Concerning Sections of Convex Sets
description: scientific article; zbMATH DE number 955340