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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Optimal-partitioning inequalities for nonatomic probability measures


Authors: John Elton, Theodore P. Hill and Robert P. Kertz
Journal: Trans. Amer. Math. Soc. 296 (1986), 703-725
MSC: Primary 60A10; Secondary 28A99, 60E15
MathSciNet review: 846603
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Abstract: Suppose $ {\mu _1}, \ldots ,{\mu _n}$ are nonatomic probability measures on the same measurable space $ (S,\mathcal{B})$. Then there exists a measurable partition $ \{ {S_i}\} _{i = 1}^n$ of $ S$ such that $ {\mu _i}({S_i}) \geq {(n + 1 - M)^{ - 1}}$ for all $ i = 1, \ldots ,n$, where $ M$ is the total mass of $ \vee _{i = 1}^n\,{\mu _i}$ (the smallest measure majorizing each $ {\mu _i}$). This inequality is the best possible for the functional $ M$, and sharpens and quantifies a well-known cake-cutting theorem of Urbanik and of Dubins and Spanier. Applications are made to $ {L_1}$-functions, discrete allocation problems, statistical decision theory, and a dual problem.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0846603-9
PII: S 0002-9947(1986)0846603-9
Keywords: Optimal-partitioning inequalities, cake-cutting, discrete allocation problems, minimax decision rules
Article copyright: © Copyright 1986 American Mathematical Society