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 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 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.

**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****5.**A. Dvoretzky, A. Wald, and J. Wolfowitz,*Relations among certain ranges of vector measures*, Pacific J. Math.**1**(1951), 59–74. MR**0043865****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

Email:
barbanej@gar.union.edu

DOI:
https://doi.org/10.1090/S0002-9939-96-03476-4

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