On the smallest disk containing a planar reduced convex body
type: Item
instance of: scholarly article
title: On the smallest disk containing a planar reduced convex body
zbMATH Open document ID: 1033.52002
published in: Archiv der Mathematik
publication date: 2003-10-16T00:00:00Z
review text: A convex body \(R\subseteq E^d\) is reduced if for every convex body \(P\subseteq R\), \(P\) different from \(R\), \(\Delta (P)< \Delta (R)\), where \(\Delta (C)\) is the thickness of the convex body \(C\subseteq E^d\), i.e. the minimum width of \(C\). The author proves that every reduced convex body \(R\subset E^2\) is contained in a disk of radius \({1\over 2}\sqrt{2}\Delta (R)\). This estimate is sharp when \(R\) is a quarter of a disk, i.e. any set \(Q\) obtained when we dissect an arbitrary disk by two perpendicular diagonals into four congruent closed sets. Moreover, the author shows that every reduced polygon \(R\) is contained in a disk of radius \({2\over 3}\Delta (R)\). This estimate is sharp only for regular triangles. It is also proved in the paper that for \(d\geq 3\) there are reduced bodies with arbitrary large finite ratio of the diameter to the thickness. This shows that there is no analogous property in \(d\geq 3\) dimensions.
reviewed by: Ferenc Fodor
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 1993488
zbMATH Keywords: Jung's theorem
zbMATH Keywords: reduced convex body
zbMATH Keywords: thickness of a convex body
zbMATH Keywords: width
author: Marek Lassak
MaRDI profile type: Publication
full work available at URL: s00013-003-4618-z
OpenAlex ID: W1968403258
DOI: 10.1007/S00013-003-4618-Z
description: scientific article; zbMATH DE number 1993488