A remark to the paper of H. Hadwiger: ``Überdeckung des Raumes durch translationsgleiche Punktmengen und Nachbarnzahl''
type: Item
instance of: scholarly article
title: A remark to the paper of H. Hadwiger: ``Überdeckung des Raumes durch translationsgleiche Punktmengen und Nachbarnzahl''
zbMATH Open document ID: 0626.10027
DOI: 10.1007/BF01326788
published in: Monatshefte für Mathematik
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178347
review text: Let \(\Lambda \in {\mathbb{R}}^ n\) be a point lattice, \(A\subset {\mathbb{R}}^ n\) be a Lebesgue-measurable set and V be the Lebesgue measure. In the paper the inequality \[ (1)\quad \alpha (A)\cdot N(A)\leq V((A-A)\cap \Lambda;A)-V(A) \] is proved, where \(\alpha\) (A) is the measure of covering \({\mathbb{R}}^ n\) by lattice translates of A, N(A) is the number of neighbours of A and \(V((A-A)\cap \Lambda;A)\) is a sort of ``measure'' of A introduced by the author. This result improves the inequality of \textit{H. Hadwiger} [ibid. 73, 213-217 (1969; Zbl 0182.557)]. Moreover, the inequality (2)\ 2(V(A)-\(\alpha\) (A))\(\leq \alpha (A)\cdot N(A)\) is also proved as a consequence of an earlier result of the author [Period. Math. Hung. 11, 95-103 (1980; Zbl 0412.10023)].
Mathematics Subject Classification ID: 11H16
Mathematics Subject Classification ID: 11H06
Mathematics Subject Classification ID: 11H31
Mathematics Subject Classification ID: 52C17
zbMATH DE Number: 4017006
zbMATH Keywords: measure of lattice covering
zbMATH Keywords: number of lattice points in difference set
zbMATH Keywords: point lattice
zbMATH Keywords: inequality
zbMATH Keywords: lattice translates
author: Béla Uhrin
MaRDI profile type: Publication
cites work: Überdeckung des Raumes durch translationsgleiche Punktmengen und Nachbarnzahl
cites work: Q5521283
description: scientific article; zbMATH DE number 4017006