Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

On the possibilities for partitioning a cake

Author(s): Julius B. Barbanel
Journal: Proc. Amer. Math. Soc. 124 (1996), 3443-3451.
MSC (1991): Primary 28A60; Secondary 90D06
MathSciNet review: 1343680
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Abstract: We wish to consider the following type of cake division problem: There are $p$ individuals. Each individual has available a measure that he or she uses to evaluate the sizes of pieces of cake. We wish to partition our cake into $q$ pieces in such a way that the various evaluations that the individuals make of the sizes of the pieces satisfy certain pre-assigned equalities and inequalities. Our main result yields a quite general criterion for showing that certain such partitions exist. Following the proof, we consider various applications.

References:

1.: J. B. Barbanel, Super envy-free cake division and independence of measures, to appear in the Journal of Mathematical Analysis and Applications 197 (1996), pp. 54--60.
2.: J. B. Barbanel and W. Zwicker, Two applications of a theorem of Dvoretsky, Wald, and Wolfovitz to cake division, pre-print.
3.: S. J. Brams and A. D. Taylor, An envy-free cake division protocol, American Mathematical Monthly 102 (1995), pp. 9--18.
4.: L. E. Dubins and E. H. Spanier, How to cut a cake fairly, American Mathematical Monthly 68 (1961), pp. 1--17. MR 23:B2068
5.: A. Dvoretsky, A. Wald, and J. Wolfovitz, Relations among certain ranges of vector measures, Pacific Journal of Mathematics 1 (1951), pp. 59--74. MR 13:331f
6.: A. Lyapounov, Sur les fonctions-vecteurs completement additives, Bull Acad. Sci. URSS 6 (1940), pp. 465--478.

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