A minimal Lebesgue covering exists
type: Item
instance of: scholarly article
title: A minimal Lebesgue covering exists
zbMATH Open document ID: 0638.52004
DOI: 10.1007/BF01142479
author: M. D. Kovalev
published in: Mathematical Notes
publication date: 1986-01-01T00:00:00Z
review text: A famous problem, due to Lebesgue, consists in determining the minimal area of covering figure \(\Phi\) for the family of all sets whose diameter is not greater than one. It is known that \(\Phi\) is not convex. A covering figure will be called irreducible, if no of its proper subsets is also a covering. The main result of the paper is the following Theorem. Any irreducible covering figure \(\Phi\) of minimal area is star-shaped; it is possible to choose the center of a polar system of coordiates inside \(\Phi\) so that its boundary \(\partial \Phi\) is given by the equation \(\tau =\rho(\phi)\), \(0\leq \phi \leq 2\pi\), where the function \(\rho(\phi)\) satisfies the inequalities: 1) \(c^ 2\leq \rho (\phi)\leq\leq 1-c^ 2;\) here \(c=1- 1/\sqrt{3},\) and 2) \(| \rho (\phi_ 1)-\rho (\phi_ 2)| \leq 5\cdot | \phi_ 1-\phi_ 2|\) for any \(\phi_ 1,\phi_ 2\in [0,2\pi]\) independent of the choice of the polar axis.
Mathematics Subject Classification ID: 52A30
Mathematics Subject Classification ID: 51M20
Mathematics Subject Classification ID: 51M25
zbMATH DE Number: 4040712
zbMATH Keywords: irreducible covering figure of minimal area
zbMATH Keywords: star-shaped
reviewed by: Valeriu Soltan
MaRDI profile type: Publication
cites work: Q5525121
cites work: Q5644548
cites work: Q5533987
cites work: Q3311230
cites work: Q3900609
full work available at URL: https://doi.org/10.1007/bf01142479
OpenAlex ID: W1998296522
description: scientific article; zbMATH DE number 4040712