Irregularities of point distributions relative to homothetic convex bodies. I
type: Item
instance of: scholarly article
title: Irregularities of point distributions relative to homothetic convex bodies. I
zbMATH Open document ID: 0841.11040
DOI: 10.1007/BF01294860
author: Gyula Károlyi
published in: Monatshefte für Mathematik
publication date: 1996-07-16T00:00:00Z
full work available at URL: https://eudml.org/doc/178705
review text: Let \(A\subset \mathbb{R}^k\) be a convex body and \(\mu\) the \(k\)-dimensional Lebesgue measure. For an arbitrary distribution \(P= \{p_1, \dots, p_N\}\) of \(N\) points in the unit cube \([0,1)^k\) let \(\Delta (P, A)= \sup|Z(P, (\lambda A+v)\cap [0,1 )^k- N\mu (\lambda A)|\), where the supremum is extended over all real numbers \(-1\leq \lambda\leq 1\) and all vectors \(v\in \mathbb{R}^k\). In the two-dimensional case \textit{J. Beck} [Proc. Lond. Math. Soc., III. Ser. 56, 1-50 (1988; Zbl 0644.10038)]\ has proved an upper bound for \(\Delta (P,A)\). In the present paper this upper bound is extended to higher dimensions. The main tool is a geometric lemma which transforms the problem to a 2-coloring problem of a hypergraph.
Mathematics Subject Classification ID: 11K38
Mathematics Subject Classification ID: 52A37
Mathematics Subject Classification ID: 05C15
Mathematics Subject Classification ID: 11K06
zbMATH DE Number: 838722
zbMATH Keywords: irregularities of point distributions
zbMATH Keywords: 2-coloring problem of a hypergraph
zbMATH Keywords: convex body
reviewed by: Robert F. Tichy
MaRDI profile type: Publication
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OpenAlex ID: W1982621289
description: scientific article; zbMATH DE number 838722