Inequalities for 𝛼-optimal partitioning of a measurable space

Legut, Jerzy

doi:10.1090/S0002-9939-1988-0969055-4

Inequalities for $\alpha$-optimal partitioning of a measurable space
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by Jerzy Legut

Proc. Amer. Math. Soc. 104 (1988), 1249-1251

DOI: https://doi.org/10.1090/S0002-9939-1988-0969055-4

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Abstract:

An $\alpha$-optimal partition $\{ A_i^ * \} _{i = 1}^n$ of a measurable space according to $n$ nonatomic probability measures $\{ {\mu _i}\} _{i = 1}^n$ is defined. A two-sided inequality for ${\nu ^ * } = \min \alpha _i^{ - 1}{\mu _i}(A_i^ * )$ is given. This estimation generalizes and improves a result of Elton et al. [3].

References

L. E. Dubins and E. H. Spanier, How to cut a cake fairly, Amer. Math. Monthly 68 (1961), 1–17. MR 129031, DOI 10.2307/2311357
A. Dvoretzky, A. Wald, and J. Wolfowitz, Relations among certain ranges of vector measures, Pacific J. Math. 1 (1951), 59–74. MR 43865, DOI 10.2140/pjm.1951.1.59
John Elton, Theodore P. Hill, and Robert P. Kertz, Optimal-partitioning inequalities for nonatomic probability measures, Trans. Amer. Math. Soc. 296 (1986), no. 2, 703–725. MR 846603, DOI 10.1090/S0002-9947-1986-0846603-9
J. Legut, Market games with a continuum of indivisible commodities, Internat. J. Game Theory 15 (1986), no. 1, 1–7. MR 839092, DOI 10.1007/BF01769272
Jerzy Legut and Maciej Wilczyński, Optimal partitioning of a measurable space, Proc. Amer. Math. Soc. 104 (1988), no. 1, 262–264. MR 958079, DOI 10.1090/S0002-9939-1988-0958079-9
H. Steinhaus, Sur la division pragmatique, Econometrica 17 (1949), no. (Sup, (Supplement), 315–319 (French). MR 39231, DOI 10.2307/1907319