A sharp nonconvexity bound for partition ranges of vector measures with atoms
type: Item
instance of: scholarly article
title: A sharp nonconvexity bound for partition ranges of vector measures with atoms
zbMATH Open document ID: 0955.28005
author: Pieter C. Allaart
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-10-25T00:00:00Z
review text: The partition range of a vector measure \(\underline\mu=(\mu_1,\dots,\mu_n)\) on a measurable space \((\Omega, {\mathcal F})\) is the subset of \({\mathbb R}^n\) given by \(R(\underline\mu)=\{(\mu_1(A_1),\dots, \mu_n(A_n)):(A_1,\dots, A_n)\) is a measurable partition of \(\Omega\}\). If each \(\mu_i\) is atomless, a result of \textit{A.~Dvoretzky, A.~Wald} and \textit{J.~Wolfowitz} [Pac. J. Math. 1, 59-74 (1951; Zbl 0044.15002)] says that \(R(\underline\mu)\) is convex and compact (a generalization of Lyapounov's Convexity Theorem). The main result in this article is a sharp upper bound for the degree of nonconvexity of \(R(\underline\mu)\) as a function of the maximum (one-dimensional) mass of the atoms of \(\underline\mu\); namely, that the Hausdorff distance (with respect to the sup norm) from \(R(\underline\mu)\) to its convex hull is at most \(\alpha (n-1)/n\), where \(\alpha\) is the size of the largest atom. This bound improves on a result in [\textit{T. P. Hill} and \textit{Y.~Tong}, Ann. Stat. 17, No. 3, 1325-1334 (1989; Zbl 0683.62036)] by an order of magnitude \(\sqrt n\). The proof is deep, and uses ideas from graph theory, combinatorics, convex geometry (Shapley-Folkman lemma), univariate optimization, and a key proportionality equation (Lemma~3.6). The main inequality is a natural generalization of the convexity theorems of Lyapounov, and of Dvoretzky, Wald and Wolfowitz, and applications to fair division problems are given, as well as generalizations, to the case of measures with atoms, of several optimal-partitioning inequalities.
Mathematics Subject Classification ID: 28B05
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 62G10
zbMATH DE Number: 1332893
zbMATH Keywords: vector measure
zbMATH Keywords: range
zbMATH Keywords: partition range
zbMATH Keywords: vector atom
zbMATH Keywords: Lyapounov convexity theorem
zbMATH Keywords: digraph
zbMATH Keywords: tree
zbMATH Keywords: fair division
zbMATH Keywords: optimal partitioning
zbMATH Keywords: cake-cutting problem
reviewed by: Theodore P. Hill
MaRDI profile type: Publication
OpenAlex ID: W2080626399
cites work: Discrete and Continuous Bang-Bang and Facial Spaces Or: Look for the Extreme Points
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cites work: A Generalization of Lyapounov's Convexity Theorem to Measures with Atoms
cites work: Optimal-Partitioning Inequalities for Nonatomic Probability Measures
cites work: A Characterization of Vector Measures with Convex Range
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cites work: Q5515142
cites work: A sharp partitioning-inequality for non-atomic probability measures based on the mass of the infimum of the measures
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cites work: Optimal-partitioning inequalities in classification and multi-hypotheses testing
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cites work: Q5779006
DOI: 10.1006/JMAA.1999.6402
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description: scientific article; zbMATH DE number 1332893