The Brunn-Minkowski inequality for volume differences
type: Item
instance of: scholarly article
title: The Brunn-Minkowski inequality for volume differences
zbMATH Open document ID: 1056.52003
DOI: 10.1016/S0196-8858(03)00095-2
published in: Advances in Applied Mathematics
publication date: 2004-10-12T00:00:00Z
review text: Suppose that \(K\), \(L\), \(D\), \(D'\) are compact domains in \(\mathbb{R}^n\) such that \(D\) and \(D'\) are homothetic and convex and \(D\subset K\), \(D'\subset L\). It is proved (in a more general form) that for the volume \(V\) one has \[ ((V(K+ L)- V(D+ D'))^{1/n}\geq (V(K)- V(D))^{1/n}+ (V(L)- V(D'))^{1/n}. \] This extends the Brunn-Minkowski inequality, and it is deduced from the latter with the help of an algebraic inequality of Bellman. Similar extensions are given for Minkowski's inequality for mixed volumes, the matrix analogue of the Brunn-Minkowski inequality, and the generalized Brunn-Minkowski inequality for quermassintegrals. Reviewer's remark: The convexity of \(D\) and \(D'\) is not assumed in the paper, but without it the employed equation \(V(D+ D')^{1/n}= V(D)^{1/n}+ V(D')^{1/n}\) is not true in general. Moreover, the equality conditions stated in Theorem 1 do not hold without additional convexity assumptions.
reviewed by: Rolf Schneider
Mathematics Subject Classification ID: 52A38
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 2107254
zbMATH Keywords: Compact domain
zbMATH Keywords: Convex body
zbMATH Keywords: Mixed volume
zbMATH Keywords: Brunn-Minkowski inequality
zbMATH Keywords: Minkowski inequality
zbMATH Keywords: Isoperimetric inequality
zbMATH Keywords: Volume difference
zbMATH Keywords: Quermassintegral difference
author: Gangsong Leng
MaRDI profile type: Publication
cites work: Q5752913
cites work: Q5336417
cites work: The Brunn-Minkowski inequality in Gauss space
cites work: Capacitary inequalities of the Brunn-Minkowski type
cites work: Q4039749
cites work: On the case of equality in the Brunn-Minkowski inequality for capacity
cites work: On the similarity of the entropy power inequality and the Brunn- Minkowski inequality (Corresp.)
cites work: Information theoretic inequalities
cites work: Q5687941
cites work: Q4934489
cites work: A Brunn-Minkowski inequality for the integer lattice
cites work: The Brunn-Minkowski inequality
cites work: A Minkowski problem for electrostatic capacity
cites work: Brunn-Minkowski inequality for multiplicities
cites work: The isoperimetric inequality
cites work: Q5287551
cites work: The affine Sobolev inequality.
description: scientific article; zbMATH DE number 2107254