The cross-section body, plane sections of convex bodies and approximation of convex bodies. II
type: Item
instance of: scholarly article
title: The cross-section body, plane sections of convex bodies and approximation of convex bodies. II
zbMATH Open document ID: 0914.52002
DOI: 10.1023/A:1004955802773
published in: Geometriae Dedicata
publication date: 1998-06-11T00:00:00Z
review text: The authors continue their investigations begun in part I of this paper [ibid. 63, No. 3, 267-296 (1996; Zbl 0865.52005)]. They first study inequalities between volumes of \(k\)-dimensional projections of convex bodies and of projections of their k-dimensional sections, as well as dual versions of such results, involving sections of the bodies and sections of circumscribed cylinders. The next topic are Minkowski sums of random segments in a convex body \(K\subset\mathbb{R}^n\). Let \(x_1,y_1, \dots, x_n\), \(y_n(n\geq d)\) be independent uniform random points in \(K\) and let \(nD\) be the Minkowski sum of the segments with endpoints \(x_i,y_i(i=1,\dots,n)\). The authors study (in more general versions) the expected volume of \(D\), and for given volume of \(K\) they find sharp lower and, if \(d = 2\), also upper bounds. The results and methods are analogous to those in earlier investigations on convex hulls of random points. The final section presents some examples related to (volume) maximal cross sections of convex bodies.
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A22
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 1161898
zbMATH Keywords: projection volume
zbMATH Keywords: section volume
zbMATH Keywords: random polytope
zbMATH Keywords: cross-section body
author: E. jun. Makai
author: Horst Martini
reviewed by: Rolf Schneider
MaRDI profile type: Publication
full work available at URL: a:1004955802773
OpenAlex ID: W4235046525
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description: scientific article; zbMATH DE number 1161898