It is always possible to "fairly" divide a cake among people using only vertical cuts. Furthermore, it is possible
to cut and divide a cake such that each person believes that everyone has
received
of the cake according to his own measure (Steinhaus 1999, pp. 65-71). Finally,
if there is some piece on which two people disagree, then there is a way of partitioning
and dividing a cake such that each participant believes that he has obtained more
than of the cake according to his own measure.
There are also similar methods of dividing collections of individually indivisible objects among two or more people when cash payments are used to even up the final division (Steinhaus 1999, pp. 67-68).
Ignoring the height of the cake, the cake-cutting problem is really a question of fairly dividing a circle into equal area pieces using cuts in its
plane. One method of proving fair cake cutting to always be possible relies on the
Frobenius-König theorem.
Beck, A. "Constructing a Fair Share." Amer. Math. Monthly94, 157-162, 1987.Brams, S. J.; Jones,
M. A.; and Klamler, C. "Better Ways to Cut a Cake." Not. Amer.
Math. Soc.53, 1314-1321, 2006.Brams, S. J. and Taylor,
A. D. "An Envy-Free Cake Division Protocol." Amer. Math. Monthly102,
9-19, 1995.Brams, S. J. and Taylor, A. D. Fair
Division: From Cake-Cutting to Dispute Resolution. New York: Cambridge University
Press, 1996.Dubbins, L. "Group Decision Devices." Amer.
Math. Monthly84, 350-356, 1997.Dubbins, L. and Spanier,
E. "How to Cut a Cake Fairly." Amer. Math. Monthly68, 1-17,
1961.Gale, D. "Dividing a Cake." Math. Intel.15,
50, 1993.Hill, T. "Determining a Fair Border." Amer. Math.
Monthly90, 438-442, 1983.Hill, T. P. "Mathematical
Devices for Getting a Fair Share." Amer. Sci.88, 325-331, Jul.-Aug.
2000.Jones, M. L. "A Note on a Cake Cutting Algorithm of Banach
and Knaster." Amer. Math. Monthly104, 353-355, 1997.Knaster,
B. "Sur le problème du partage pragmatique de H. Steinhaus."
Ann. de la Soc. Polonaise de Math.19, 228-230, 1946.Rebman,
K. "How to Get (At Least) a Fair Share of the Cake." In Mathematical
Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 22-37,
1979.Robertson, J. and Webb, W. Cake
Cutting Algorithms: Be Fair If You Can. Wellesley, MA: A K Peters, 1998.Steinhaus,
H. "Remarques sur le partage pragmatique." Ann. de la Soc. Polonaise
de Math.19, 230-231, 1946.Steinhaus, H. "The Problem
of Fair Division." Econometrica16, 101-104, 1948.Steinhaus,
H. "Sur la division pragmatique." Ekonometrika (Supp.)17,
315-319, 1949.Steinhaus, H. Mathematical
Snapshots, 3rd ed. New York: Dover, pp. 64-67, 1999.Stromquist,
W. "How to Cut a Cake Fairly." Amer. Math. Monthly87, 640-644,
1980.
people using only vertical cuts. Furthermore, it is possible
to cut and divide a cake such that each person believes that everyone has
received 
of the cake according to his own measure (Steinhaus 1999, pp. 65-71). Finally,
if there is some piece on which two people disagree, then there is a way of partitioning
and dividing a cake such that each participant believes that he has obtained more
than 
of the cake according to his own measure.
equal area pieces using cuts in its
plane. One method of proving fair cake cutting to always be possible relies on the
Frobenius-König theorem.
