Note on an inequality of Wegner
type: Item
instance of: scholarly article
title: Note on an inequality of Wegner
zbMATH Open document ID: 1117.52020
author: Károly jun. Böröczky
author: Imre Z. Ruzsa
published in: Discrete \& Computational Geometry
publication date: 2007-04-26T00:00:00Z
review text: For \(n\geq 2\) non-overlapping unit discs in the Euclidean plane the authors study the minimum area of their convex hull. More precisely, if \(D_n\) denotes this convex hull and \(A(D_n)\) its area, the authors derive an upper bound on \(A(D_n)\) reached by a so-called Groemer packing, namely \[ A(D_n)\leq 2\sqrt{3}\cdot(n- 1)+ (2- \sqrt{4})\cdot \lceil\sqrt{12n-3}- 2\rceil+ \pi. \] They also give a number theoretic characterization of all \(n\) such that a so-called Wegner packing (which is given if and only if the lower bound on \(A(D_n)\) is attained) of \(n\) unit circles exists. They show that these numbers constitute \(95,83\dots\%\) of \(\mathbb{N}\). Finally an interesting conjecture is posed, referring to the minimum of \(A(D_n)\) in view of the perimeter of \(D_n\).
reviewed by: Horst Martini
Mathematics Subject Classification ID: 52C15
zbMATH DE Number: 5146114
zbMATH Keywords: Wegner packing
zbMATH Keywords: Groemer packing
zbMATH Keywords: Oler inequality
zbMATH Keywords: Thue-Groemer inequality
zbMATH Keywords: convex hull
zbMATH Keywords: packing inequality
MaRDI profile type: Publication
full work available at URL: s00454-006-1277-4
OpenAlex ID: W2035244142
DOI: 10.1007/S00454-006-1277-4
description: scientific article; zbMATH DE number 5146114