On a problem of L. Fejes Toth
type: Item
instance of: scholarly article
title: On a problem of L. Fejes Toth
zbMATH Open document ID: 0444.52008
DOI: 10.1016/0012-365X(80)90111-9
author: János Pach
author: P. Erdős
published in: Discrete Mathematics
publication date: 1980-01-01T00:00:00Z
review text: Let \(0\leq x_1\leq x_2\leq x_3\leq\dots\) be a sequence of real numbers, \(\lim x_i = +\infty\). The authors prove that if \(\sum_il/x_i^{n-k}=+\infty\) then there exists a point-system \(P=\{z_1,z_2,\dots\}\) in the \(n\)-dimensional space \(\mathbb E^n\), for which \(|z_i|=x_i\) holds (\(i=1,2,\dots\)), and any \(k\)-dimensional plane comes arbitrarily near to \(P\). this result is best possible in the sense that if \(P\{z_1,z_2,\dots\}\) is a point-system satisfying \(\sum_il/|z_i|^{n-k}< +\infty\) then for every \(C> 0\) there exists a \(k\)-dimensional plane in \(BbbE^n\), whose distance from all members of \(P\) is at least \(C\). A generalization is also proved. This settles a problem of \textit{L. Fejes Tóth} [Mat. Lapok 25 (1974), 13-20 (1976; Zbl 0359.52010)].
Mathematics Subject Classification ID: 52A37
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 3694086
zbMATH Keywords: countable point-system in \(E^2\)
zbMATH Keywords: plane comes arbitrarily near to \(P\)
Wikidata QID: Q29039011
MaRDI profile type: Publication
cites work: Q4131254
cites work: Q5845761
full work available at URL: 0012-365x(80)90111-9
OpenAlex ID: W2088558457
DBLP publication ID: journals/dm/ErdosP80
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description: scientific article; zbMATH DE number 3694086