The Pentagonal Pizza Conjecture

Abstract

The pentagonal pizza conjecture says that it is impossible to cut a pentagonal pizza into eight slices by four concurrent straight lines that make four equal smaller angles alternating with four equal bigger angles, and such that all smaller angle slices have the same area and all bigger angle slices have the same area. The conjecture is proved to be true when the pizza is shaped as an ellipse, triangle, or quadrilateral, except for a square, and is proved not true for all n-gons, where $n\ge 6$. These results are closely related to the first theorem of convex geometry, Zindler’s theorem of 1920. Similar results are proved for perimeter instead of area.

MSC:

• Primary 52A38
• Secondary
• 51N20
• 52A10
• 53A04

Acknowledgments

We thank the editors and the anonymous reviewers for their support, careful reading, and suggestions for improvement. Stefan Catoiu gratefully acknowledges support from the University Research Council Faculty Paid Leave Program at DePaul University.

Additional information

Notes on contributors

Allan Berele

Allan Berele received his Ph.D. from the University of Chicago under the direction of I. N. Herstein, with A. Regev as his unofficial adviser, which is why his main research interest is in rings with polynomial identities. When Allan Berele was much younger he found an amazing book in a drugstore called College Geometry, by Nathan Altshiller-Court. It had a big influence on him which he still feels today.

Stefan Catoiu

Stefan Catoiu received his undergraduate and master’s degrees from the University of Bucharest, and Ph.D. from the University of Wisconsin-Madison under the direction of Donald S. Passman. He held a visiting position at Temple University before joining the faculty at DePaul University. His research interests include noncommutative algebra, real analysis, geometry, number theory, and elementary mathematics.