The Brunn-Minkowski inequality, Minkowski's first inequality, and their duals
type: Item
instance of: scholarly article
title: The Brunn-Minkowski inequality, Minkowski's first inequality, and their duals
zbMATH Open document ID: 0959.52007
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-05-01T00:00:00Z
review text: Let \(K,L\) be convex bodies in Euclidean space \(\mathbb{E}^n\) with volumes \(V(K)=V(L)=1\), and let \(V_1(K,L)\) denote the mixed volume \(V(K, \dots, K,L)\). Then \[ V(K+L)^{1/n} -2\leq V_1(K,L) -1\leq {1\over n}\bigl(V(K+L)-2^n \bigr). \] These inequalities provide a quantitative improvement of the known equivalence of the Brunn-Minkowski inequality and Minkowski's first inequality for convex bodies. The left-hand inequality is due to \textit{H. Groemer} [Geom. Dedicata 33, No.~1, 117-122 (1990; Zbl 0692.52005)], the right-hand one is proved here, together with a variant and analogues for star bodies in Lutwak's dual Brunn-Minkowski theory. The main results are shown to be best possible in a precise sense.
reviewed by: Rolf Schneider
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 1471627
zbMATH Keywords: Brunn-Minkowski theorem
zbMATH Keywords: Minkowski's first inequality
zbMATH Keywords: dual Brunn-Minkowski theory
author: Richard J. Gardner
author: Salvatore Vassallo
MaRDI profile type: Publication
full work available at URL: jmaa.2000.6774
OpenAlex ID: W1987697227
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DOI: 10.1006/JMAA.2000.6774
description: scientific article; zbMATH DE number 1471627