Some Recent Results on Pie Cutting

Abstract

For cake cutting, cuts are parallel to an axis and yield rectangular pieces.  As such, cutting a cake is viewed as dividing a line segment.  For pie cutting, cuts are radial from the center of a disc to the circumference and yield sectors or wedge-shaped pieces.  As such, cutting a pie is viewed as dividing a circle.  There is clearly a relationship between cutting a cake and cutting a pie.  Once a circular pie has a single cut, then it can be straightened out into a segment, looking like a cake.  Isn't a cake just a pie that has been cut?  Gale (1993) suggested that this topology was a significant difference.  This note is to summarize and compare some of the recent results on pie cutting that appear in Barbanel and Brams (2007) and Brams, Jones, and Klamler (2007).  The geometric framework presented in Barbanel and Brams (2007) is used to prove and to explain results in Brams, Jones, and Klamler (2007).