Some Recent Results on Pie Cutting

For cake cutting, cuts are parallel to an axis or segment 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. This difference in topology is apparent in the minimal number of cuts necessary to divide the desserts. For n players, it takes a minimal number of n− 1 cuts to divide a linear cake and n cuts to divide a circular pie so that each player receives a single piece. Though, 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). In particular, the geometric structure presented in Barbanel and Brams (2007) is used to prove and to explain results in Brams, Jones, and Klamler (2007). Pie cutting or dividing a circle is more appropriate for some applications. For example, dividing a shoreline between more than two vendors would result in disconnected shorelines for a vendor if a cake-cutting procedure were to be used (see Figure 1). The circular structure is also useful when staffing employees (e.g., emergency room physicians, police, etc.) or allocating machine time (e.g., production plants, computer time, etc.) over 24-hour periods.