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Transactions of the American Mathematical Society

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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.

References [Enhancements On Off] (What's this?)

  • [1] R. Ash, Real analysis and probability, Academic Press, New York, 1972. MR 0435320 (55:8280)
  • [2] S. Demko and T. Hill, Equitable distribution of indivisible objects, preprint, 1985. MR 964811 (89j:90005)
  • [3] L. Dor, On projections in $ {L_1}$, Ann. of Math (2)102 (1975), 463-474. MR 0420244 (54:8258)
  • [4] L. Dubins and E. Spanier, How to cut a cake fairly, Amer. Math. Monthly 68 (1961), 1-17. MR 0129031 (23:B2068)
  • [5] N. Dunford and J. Schwartz, Linear operators, Part I, Interscience, New York, 1958.
  • [6] A. Dvoretzky, A. Wald and J. Wolfowitz, Relations among certain ranges of vector measures, Pacific J. Math. 1 (1951), 59-74. MR 0043865 (13:331f)
  • [7] T. Hill, Equipartitioning the common domain of nonatomic measures, Math. Z. 189 (1985), 415-419. MR 783565 (86e:28005)
  • [8] J. Stoer and C. Witzgall, Convexity and optimization in finite dimensions. I, Grundlehren Math. Wiss., Band 163, Springer-Verlag, New York, 1970. MR 0286498 (44:3707)
  • [9] K. Urbanik, Quelques théorèmes sur les mesures, Fund. Math. 41 (1955), 150-162. MR 0063427 (16:120d)

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Keywords: Optimal-partitioning inequalities, cake-cutting, discrete allocation problems, minimax decision rules
Article copyright: © Copyright 1986 American Mathematical Society

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