An extension of Jung's theorem
type: Item
instance of: scholarly article
title: An extension of Jung's theorem
zbMATH Open document ID: 0574.52012
DOI: 10.1007/BF02761397
published in: Israel Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: Jung's theorem is related to Helly's theorem and can be stated as follows: Let \(C\subset {\mathbb{R}}^ n\) be a compact convex set of unit circumradius and let B be the unit ball containing C. Then diam C is not less than \(\sqrt{(2n+2)/n}\) (which is the edge length of a regular n- simplex s inscribed in B). If diam C\(=\sqrt{(2n+2)/n}\) then C contains such a simplex s. The author's extension of Jung's theorem answers the following questions. Suppose diam C is greater than but close to \(\sqrt{(2n+2)/n}\). Then C should contain an n-simplex s' inscribed in B and close to s. How close? What is the range of edge length of s' in terms of the diameter and dimension of C? Lower bounds for the inradius and minimal width are also obtained. The proofs rely on Caratheodory theorem (If \(X\subset {\mathbb{R}}^ n\), each point of the convex hull of X is a convex combination of at most \(n+1\) points of X''), induction on the dimension and the study of some properties of simplices.
Mathematics Subject Classification ID: 52A35
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 3917141
zbMATH Keywords: Helly type theorem
zbMATH Keywords: theorem of Jung
zbMATH Keywords: convex set
zbMATH Keywords: circumradius
zbMATH Keywords: diameter
zbMATH Keywords: inradius
zbMATH Keywords: minimal width
zbMATH Keywords: simplices
author: B. V. Dekster
reviewed by: Francisco Bernis
MaRDI profile type: Publication
cites work: Q5556250
cites work: Q5547252
cites work: Two inequalities for convex sets in the plane
cites work: A family of inequalities for convex sets
cites work: SETS OF CONSTANT WIDTH AND INEQUALITIES
full work available at URL: https://doi.org/10.1007/bf02761397
OpenAlex ID: W2355446191
description: scientific article; zbMATH DE number 3917141