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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the possibilities for partitioning a cake
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by Julius B. Barbanel
Proc. Amer. Math. Soc. 124 (1996), 3443-3451
DOI: https://doi.org/10.1090/S0002-9939-96-03476-4
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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
  • 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.
  • J. B. Barbanel and W. Zwicker, Two applications of a theorem of Dvoretsky, Wald, and Wolfovitz to cake division, pre-print.
  • S. J. Brams and A. D. Taylor, An envy-free cake division protocol, American Mathematical Monthly 102 (1995), pp. 9–18.
  • 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
  • C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
  • A. Lyapounov, Sur les fonctions-vecteurs completement additives, Bull Acad. Sci. URSS 6 (1940), pp. 465–478.
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Bibliographic Information
  • Julius B. Barbanel
  • Affiliation: Department of Mathematics, Union College, Schenectady, New York 12308
  • Email: barbanej@gar.union.edu
  • Received by editor(s): October 3, 1994
  • Received by editor(s) in revised form: May 19, 1995
  • Communicated by: Andreas R. Blass
  • © Copyright 1996 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 124 (1996), 3443-3451
  • MSC (1991): Primary 28A60; Secondary 90D06
  • DOI: https://doi.org/10.1090/S0002-9939-96-03476-4
  • MathSciNet review: 1343680