Balanced convex partitions of measures in \(\mathbb R^{2}\)
type: Item
instance of: scholarly article
title: Balanced convex partitions of measures in \(\mathbb R^{2}\)
zbMATH Open document ID: 1002.52002
author: Toshinori Sakai
published in: Graphs and Combinatorics
publication date: 2002-05-14T00:00:00Z
review text: Let \(n \geq 2\) be an integer and let \(\mu_1\) and \(\mu_2\) be measures in \(R^2\) such that each \(\mu_i\) is absolutely continuous with respect to the Lebesgue measure and \(\mu_1(R^2)=\mu_2(R^2)=n\). The author shows that if \(\mu_1(B)=\mu_2(B)=n\) for some bounded domain \(B\), then there exist positive integers \(n_1\), \(n_2\) with \(n_1+n_2=n\) and disjoint open halfplanes \(D_1\), \(D_2\) such that the closure of \(D_1 \cup D_2\) is \(R^2\), \(\mu_1(D_1)= \mu_2(D_1) = n_1\) and \(\mu_1(D_2)= \mu_2(D_2) = n_2\); or there exist positive integers \(n_1\), \(n_2\), \(n_3\) with \(n_1+n_2+n_3=n\) and disjoint open convex domains \(D_1\), \(D_2\), \(D_3\) such that the closure of \(D_1 \cup D_2 \cup D_3\) is \(R^2\), \(\mu_1(D_1)= \mu_2(D_1) = n_1\), \(\mu_1(D_2)= \mu_2(D_2) = n_2\), \(\mu_1(D_3)= \mu_2(D_3) = n_3\) and such that the ray \(closure(D_1) \cap closure(D_2)\) is parallel to any non-null vector in \(R^2\). He also shows a similar result for partitions of point sets on the plane. As a consequence of both results he obtains several corollaries giving sufficient conditions for the existence of \(k\)-convex partitions and 3-radial convex partitions.
Mathematics Subject Classification ID: 52A10
zbMATH DE Number: 1740500
zbMATH Keywords: partitions
zbMATH Keywords: \(k\)-convex partitions
zbMATH Keywords: \(k\)-radial partitions
zbMATH Keywords: absolute continuous measures
MaRDI profile type: Publication
full work available at URL: s003730200011
OpenAlex ID: W2092399768
DOI: 10.1007/S003730200011
description: scientific article; zbMATH DE number 1740500