A generalization of Jung's theorem
type: Item
instance of: scholarly article
title: A generalization of Jung's theorem
zbMATH Open document ID: 0759.52008
DOI: 10.1007/BF00147552
published in: Geometriae Dedicata
publication date: 1992-09-27T00:00:00Z
review text: In 1901 \textit{H. W. E. Jung} had established a relation between the circumradius and the diameter of a convex body [J. Reine Angew. Math. 123, 241-257 (1901)]. The diameter of a convex body can be interpreted as the maximum of the circumradii of all 1-dimensional sections or 1- dimensional orthogonal projections of that body. Therefore the author suggests two kinds of i-dimensional circumradii of a \(d\)-dimensional convex body (\(1\leq i\leq d\)), using the section or the projection view. For these circumradii of a convex body he establishes two inequalities between circumradii of dimensions \(i\) und \(j\) (\(1\leq j\leq i\leq d\)) with equality exactly for a regular simplex. His inequalities can be interpreted as a generalization of the classical theorem of Jung, namely for \(i=j=1\) both inequalities give Jung's theorem. To prove these theorems it suffices to consider simplices since the circumradius of a convex body is determined by the circumradius of a certain simplex belonging to that body. In a similar way one can define series of inradii. Some of these functionals are studied by \textit{P. Gritzmann} and \textit{V. Klee} [Discrete Comput. Geom. 7, No. 3, 255-280 (1992)].
reviewed by: Johannes Böhm
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 57586
zbMATH Keywords: convex sets
zbMATH Keywords: diameter
zbMATH Keywords: circumradius
zbMATH Keywords: inequalities
zbMATH Keywords: \(n\)-dimensional Euclidean geometry
author: Martin Henk
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00147552
OpenAlex ID: W2088836135
description: scientific article; zbMATH DE number 57586