# Convex 'Fair' Partitions Of Convex Polygons

**Basic Question:** Given any positive integer *n*, can any convex polygon be partitioned into *n* convex pieces so that all pieces have the same area and same perimeter?

**Definitions:** Define a *Fair Partition* of a polygon as a partition of it into a finite number of pieces so that every piece has both the same area and the same perimeter. Further, if all the resulting pieces are convex, call it a *Convex Fair Partition*.

**Questions:** 1. (Rephrasing the above 'basic' question) Given any positive integer *n*, can any convex polygon be convex fair partitioned into n pieces?

2. If the answer to the above is *"Not always''*, how does one decide the possibility of such a partition for a given convex polygon and a given *n*? And if fair convex partition is allowed by a specific convex polygon for a give *n*, how does one find the *optimal* convex fair partition that *minimizes* the total length of the cut segments?

3. Finally, what could one say about *higher dimensional analogs* of this question?

**Conjecture:** The authors tend to believe that the answer to the above 'basic' question is "yes". In other words they guess: *Every* convex polygon allows a convex fair partition into *n* pieces for any *n*

1. The above conjecture is easily seen to hold for *n*=2. for *n*=3 and above, it is not clear.

2. The *n* = 2 case *does not* appear to allow a recursive generalization for values of *n* equal to powers of 2.

3. It can be shown that any polygon (not necessarily convex) allows a fair partitioning into *n* pieces for any *n*, provided the pieces need not be convex (this is not a *convex* fair partition). See (4) in references below.

4. It appears that the fair parition of a convex polygon which minimizes the total length of cuts (or equivalently, the sum of the perimeters of the pieces) *need not* be a convex fair partition.

5. There is no known work in this specific area. The problem of partitioning convex polygons into equal area convex pieces so that every piece *equally shares the boundary* of the input polygon has been studied (references below)

## Bibliography

(*)1. The original 'mainstream' statement of this problem: http://maven.smith.edu/~orourke/TOPP/P67.html#Problem.67

2. Jin Akiyama, A. Kaneko, M. Kano, Gisaku Nakamura, Eduardo Rivera-Campo, S. Tokunaga, and Jorge Urrutia. Radial perfect partitions of convex sets in the plane. In Japan Conf. Discrete Comput. Geom., pages 1-13, 1998.

3. Jin Akiyama, Gisaku Nakamura, Eduardo Rivera-Campo, and Jorge Urrutia. Perfect divisions of a cake. In Proc. Canad. Conf. Comput. Geom., pages 114-115, 1998.

4. This blog maintained by the authors has tentative thoughts, examples, etc on 'Fair Partitions': http://nandacumar.blogspot.com

* indicates original appearance(s) of problem.