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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


On the possibilities for partitioning a cake

Author: 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 | References | Similar Articles | Additional Information

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

  • 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, Amer. Math. Monthly 68 (1961), 1–17. MR 0129031 (23 #B2068)
  • 5. 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)
  • 6. A. Lyapounov, Sur les fonctions-vecteurs completement additives, Bull Acad. Sci. URSS 6 (1940), pp. 465--478.

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Additional Information

Julius B. Barbanel
Affiliation: Department of Mathematics, Union College, Schenectady, New York 12308

PII: S 0002-9939(96)03476-4
Received by editor(s): October 3, 1994
Received by editor(s) in revised form: May 19, 1995
Communicated by: Andreas R. Blass
Article copyright: © Copyright 1996 American Mathematical Society

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