Thinnest covering of the Euclidean plane with incongruent circles
type: Item
instance of: scholarly article
title: Thinnest covering of the Euclidean plane with incongruent circles
zbMATH Open document ID: 1386.52018
author: Dietmar Dorninger
published in: Analysis and Geometry in Metric Spaces
publication date: 2018-01-12T00:00:00Z
review text: For \(0 < q \leq 1\), let \(D(q)\) denote the smallest (lower) density of a covering of the Euclidean plane by circles with radii from the interval \([q,1]\). \textit{L. Fejes Tóth} and \textit{J. Molnár} conjectured that \[ D(q) \geq S(q):= \min\limits_{0 < x \leq 1} \frac{\pi-2\left(1-q^2\right)\arctan\left(\sqrt{1-x^2}/(x+q)\right)}{2\sqrt{1-x^2}(x+q)} \] [Math. Nachr. 18, 235--243 (1958; Zbl 0083.38501)]. In the paper under review that conjecture was made accessible to a computer-aided numerical confirmation. Moreover, bounds of \(S(q)\) are given.
reviewed by: Christian Richter
Mathematics Subject Classification ID: 52C15
zbMATH DE Number: 6825494
zbMATH Keywords: circular discs
zbMATH Keywords: covering of the plane
zbMATH Keywords: minimum density
zbMATH Keywords: numerical verification
zbMATH Keywords: upper and lower bounds
MaRDI profile type: Publication
full work available at URL: agms-2017-0002
OpenAlex ID: W2607457343
cites work: Q5637537
cites work: Q3282413
cites work: Q5525870
cites work: Q3278742
cites work: Überdeckung der Ebene durch inkongruente Kreise
cites work: Covering the plane with two kinds of circles
cites work: Unterdeckung und Überdeckung der Ebene durch Kreise Dem Andenken an Professor H. L. Schmid gewidmet
cites work: Covering of the plane by circles
cites work: Zum Problem der dünnsten Kreisüberdeckung der Ebene
cites work: Überdeckung der Ebene durch inkongruente Kreise. (Eine Bemerkung zu einer Arbeit von Fejes Toth und Molnar)
DOI: 10.1515/AGMS-2017-0002
description: scientific article; zbMATH DE number 6825494