Clouds of planar convex bodies
type: Item
instance of: scholarly article
title: Clouds of planar convex bodies
zbMATH Open document ID: 1001.52002
author: Lásló Szabó
author: Zoltán Ujváry-Menyhárt
published in: Aequationes Mathematicae
publication date: 2002-06-27T00:00:00Z
review text: A deep cloud is a family of convex bodies \(K_1,\dots, K_n\) arranged around a given convex body \(K\) such that the bodies \(K_1,\dots, K_n\) are pairwise non-overlapping and each ray, emanating from \(K\), intersects some of the \(K_i\) at its interior. The authors are concerned with the minimum number \(C(K)\) of congruent copies of a given convex body \(K\) in the plane yielding a deep cloud around \(K\). After shortly confirming \(C(K)\geq 4\) (with various cases of equality), they prove that \(C(K)\leq 9\), with equality iff \(K\) is a circle. Furthermore, they show that \(C(K)= 8\) if there are two concentric circles \(C_1\), \(C_2\) of radius \(r_1= 0,99\) and \(r_2= 1\), respectively, such that \(C_1\subseteq K\subseteq C_2\).
reviewed by: Horst Martini
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 52C15
zbMATH DE Number: 1761060
zbMATH Keywords: deep cloud
zbMATH Keywords: finite packings
MaRDI profile type: Publication
full work available at URL: s00010-002-8025-5
OpenAlex ID: W2004140962
DOI: 10.1007/S00010-002-8025-5
description: scientific article; zbMATH DE number 1761060